Convexity and related combinatorial geometry: Proceedings of the Second University of Oklahoma Conference 0824712781

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volume 78 ,

mathematic-

convexity and related combinatorial geometry Edited by

David (3. Kay Marilyn Breen

CONVEXITY AND RELATED COM BINATORIAL GEOMETRY

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes Executive Editor: Earl J. Taft

Edwin Hewitt University of Washington

Rutgers University New Brunswick. New Jersey

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Chairman of the Editorial Board S. Kobayashi University of California, Berkeley Berkeley. California

Editorial Board Glen E. Bredon Rutgers University

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University of Chicago

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Contributions to Leclurr Nuler in Pure and Applied Mathematics are reproduced by direct photography of the author's typewritten manuscript. Potential authors are advised to submit preliminary manuscripts for review purposes. After acceptance, the author is responsible for preparing the final manuscript in camera-ready form. suitable for direct reproduction. Marcel Dekker. inc. will furnish instructions to authors and special typing paper. Sample pages are reviewed and returned with our suggestions to assure quality control and the most attractive rendering of your manuscript The publisher will also be happy to supervise and assist in all stages ot the preparation of your camera-ready manuscript.

LECTURE NOTES

IN PURE AND APPLIED MATHEMATICS l. 2. 3. 4.

N. Jacobron, Exceptional Lie Algebras L.-A. Lindahl and F. Poulren, Thin Sets in Harmonic Analysis I. Snake, Classification Theory of Semi-Simple Algebraic Groups F. Hirzebruch, W. D. Newman, and S. S. Koh, Differentiable Manifolds and Quadratic Forms (out of print) 5. 1. Chan], Riemannian Symmetric Spaces of Rank One (out of print) 6. R. B. Burckel, Characterization of COD Among Its Subalgebras 7. B. R. McDonald, A. R. Magid, and K. C. Smith. Ring Theory: Proceedings of the

Oklahoma Conference

‘

8. Y.«T. Siu, Techniques of Extension of Analytic Objects 9. S R. Camdur, W. E. Pfaffenberger, and B. Yood, Calkin Algebras and Algebras of Operators on Banach Spaces 10. E. 0. Roxin, R-T. Liu, and R. L. Sternberg, Differential Games and Control Theory 11. M. Orzech and C. Small, The Brauer Group of Commutative Rings 12. S. Thorneier, Topology and Its Applications 13. J. M. La'pez and K. A. Roar. Sidon Sets 14. W. W. Comfort and S. Negreporm‘r, Continuous Pseudometrics 15. K. McKenrwn and .l. M. Robertson, Locally Convex Spaces [6. M. Carmeli and S. Malin, Representations of the Rotation and Lorentz Groups: An Introduction 17. G. B. Seligrrmn, Rational Methods in Lie Algebras 18. D. G. de Figueiredo, Functional Analysis: Proceedings of the Brazilian Mathematical Society Symposium 19. L. Cerari, R. Karman, and .I. D. Schuur, Nonlinear Functional Analysis and Differential

'” 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.

'

E‘-

"

of the‘" “

State U '

' yl‘

"

J. J. Scha'ffer, Geometry of Spheres in Nonned Spaces K. Yana and M. Kan, Anti-Invariant Submanifolds W. V. Varconcelus, The Rings of Dimension Two R. E. Chandler, Hausdorff Compactifications S. P. Franklin and B. V. S. Thomas, Topology: Proceedings of the Memphis State University Conference S. K. Jain, Ring Theory: Proceedings of the Ohio University Conference B. R. McDonald and R. A. Mom'x, Ring Theory [1: Proceedings of the Second Oklahoma Conference R. B. Mum and A. Rhemtulla, Orderable Groups J. R. Graef, Stability of Dynamical Systems: Theory and Applications H.-C. Wang, Homogeneous Banach Algebras E. 0. Roxin. P.-T. Liu, and R. L. Stemberg, Differential Games and Control Theory 11 R. D. I’urter, Introduction to Fibre Bundles M. Altman. Contractors and Contractor Directions Theory and Applications J. S. Golan. Decomposition and Dimension in Module Categories G. Fairweather, Finite Element Galerlrin Methods for Differential Equations I. D. Sally, Numbers of Generators of [deals in Local Rings S. S. Miller, Complex Analysis: Proceedings of the S.U.N.Y. Brockport Conference R. Gordon, Representation Theorynf Algebras: Proceedings of the Philadelphia Conference M. Gala and F. D. Grantham, Semisimple Lie Algebras A. l. Arruda. N. C. A. (in Costa, and R. Chaaqui. Mathematical Logic: Proceedings of the First Brazilian Canference I". Van Oyrtaeyen, Ring Theory: Proceeding of the 1977 Antwerp Conference

_.__D‘__ .A.-_.____ __ “AM___.‘-

Other Volume: in Preparation

gen

4 . F. Van Dyrraeyen and A. Verrchoren, Reflectors and Localization: Application to Sheaf Theory 42. M. Saryanarayana, Positively Ordered Semigroups 43. D. L. Rune", Mathematics of Finite—Dimensional Control Systems 44. P.-T. Lia and E. Raxin, Differential Games and Control Theory 111: Proceedings of the Third Kinpton Conference, Part A 45. A. Geramita and J. Seberry, Orthogonal Designs: Quadratic Forms and Hadamard Matrices 46. J. Cigler, V. Lore", and P. Michor, Banach Modules and Functors on Categories of Banach Spaces 47. P.-T. Liu and J. G. Surinen, Control Theory in Mathematical Economics: Proceedings of the Third Kingston Conference, Part B 48. C. Byrnes, Partial Differential Equations and Geometry 49. G. Klambauer, Problems and Propositions‘in Analysis 50. .I. Knap/mach" Analytic Arithmetic of Algebraic Function Fields 51. F. Van Oyrtaeyen, Ring Theory. Proceedings of the 1978 Antwerp Conference 52. B. Kedem, Binary Time Series 53. J. Barres-Nero and R. A. Arrino, Hypoelliptic Boundary-Value Problems 54. R. L. Sternberg, A. l. Kalinawrki, and]. S. Papadakir, Nonlinear Partial Differential Equations in Engineering and Applied Science SS. B. R. McDonald, Ring Theory and Algebra Ill: Proceedings of The Third Oklahoma Conference 56. I. S. Galan, Structure Sheaves over a Noncommutative Ring 57. T. V. Narayana, I. G. Williamr, and R. M. Mathren, Combinatorics, Representation Theory and Statistical Methods in Groups: YOUNG DAY Proceedings 58. T. A. Burton, Modeling and Differential Equations in Biology 59. K. H. Kim and F. W. Rourh. Introduction to Mathematical Consensus Theory 60. J. Banar and K. Goebel, Measures of Noncompactness in Banach Spaces 61. 0. A. Nielsen, Direct integral Theory 62. J. E. Smith, G. 0. Kenny, and R. N. Ball, Ordered Groups: Proceedings of the Boise State Conference 63. I. Cronin, Mathematics of Cell Electrophysiology 64,. .I. W. Brewer, Power Series Over Commutative Rings 65. P. K. Kamrhan and M. Gupta, Sequence Spaces and Series 66. T. G. McLaughlin.»Regressive Sets and the Theory of lsols (in press) 67. 7'. L. Henlman, S. M. Rankin, III, and H. W. Slrch, Integral and Functional Differential Equations 68. R. Draper. Commutative Algebra: Analytic Methods W. G. McKay and J. Patna. Tables of Dimensions, lndices, and Branching Rules 69. for Representations of Simple Lie Algebras 70. R. L. Devaney and Z. H. Nitecki, Classical Mechanics and Dynamical Systems . J. Van Gee], Places and Valuations in Noncommutative Ring Theory 72. C. Faith, lnjective Modules and Injecttve Quotient Rings 73. A. Fiacco, Mathematical Programming with Data Perturbations l 74. P. Schultz, C. Praeger. and R. Sullivan, Algebraic Structures and Applications: Proceedings of the First Western Australian Conference on Algebra 75. I.. Bican, T. Kepkn, and P. Ne’mec, Rings. Modules, and Preradicals 76. D. C. Kay and M. Breen, Convexity and Related Combinatorial Geometry: Proceedings of the Second University of Oklahoma Conference

CONVEXITY AND RELATED COM BINATORIAL GEOMETRY Proceedings of the Second University of Oklahoma Conference

edited by

David C. Kay and Marilyn Breen Department of Mathematics University of Oklahoma Norman, Oklahoma

MARCEL DEKKER, INC

New York and Basel

Library of Congress Cataloging in Publication Date Main entry under title: Convuity and related combinatorial geometry. (Lecture notes in pure and applied mathematics ; v. 76) Includes index. 1. Convex palyhedra——Cangresses. 2. Combinatorial geometry—Congresses. I. Kay, David C., [date]. II. Breen, Marilyn, [date]. 111. Series. QA6140.3.C66 516.3'6 82-1381 ISBN 0-8247-1278—1 AACRZ

COPYRIGHT © 1982 byVMARCm. DEKKER, INC.

ALL RIGHTS RESERVED

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage

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MARCH. DEICKER, INC. 270 Madison Avenue, New York, New York Current printing (last digit): 10 9 3 2 PRINTED IN THE UNITED STATES OF AMERICA

10016

PREFACE

The contents of this work consist of the papers presented at a convexity conference held at the University of Oklahoma, March l3-lS, 1980.

The parti—

cipants were particularly fortunate to hear the presentations of results and surveys by the keynote speakers Victor Klee, David Barnette, Don Chakerian, Tom Sallee, and John Reay (whose papers appear in these pro— ceedings).

All the papers presented were worthwhile and interesting, but

particularly noteworthy were (1) the outstanding survey of the d-step conjecture presented by Victor Klee (approached from an ad hoc point of view to promote the breaking away from traditional and unfruitful attacks on the problem), (2) the solution by Carl Lee of the longstanding characterization problem of the family of f—vectors of simplicial polytopes in Rd via the

Dehn—Sommerville equations (one half of the characterization--the necessity of the conditions——was accomplished earlier by R.

Stanley, referenced in

Lee's work), and (3) a very useful organization of the multitude of scattered results and ideas of abstract convexity and some new relationships in that field by Robert Jamison-Waldner.

I

All the papers appearing here have been accepted prima-facie without the aid of referees, so some authors may have published these results in a different form in referreed journals. In the opinion of its participants, the conference was an important and delightful success, highlighted by a telegram of good wishes from Ger— many by colleagues in the field, Ludwig Danzer, J. Eckoff, U. Wegner and C. Zsmfirescu. Gratitude is sent to our benefactors, without whose support the confer— ence could not have materialized:

The J. Clarence Karcher Foundation at the

University of Oklahoma which provided the major funding for the conference, and the National Science Foundation which provided extra participant sup— port.

Thanks go also to the expert job by the typista, Trish Abolins and

Detia Roe. David C. Kay Marilyn Breen

iii 6M9 b lac ks;

CONTENTS

Preface Contributors How Many Steps?

Victor Klee Polyhedral flaps on 2—manifolds

David Barnette Characterizing the Numbers 'of Faces of a Simplicial Convex Polytope Car-Z W. Lee

21

A Dual Proof of the Upper Bounn Theorem Arne ted

39

Euler's Theorem and Where It Led G. Thomas SaZZee

105

Mixed Volumes and Geometric Inequalities G. D. Chakerian

57

Intersections of Convex Sets and Surfaces Paul Goodey

63

Nonnegstive, Motion-invariant Valuations of Convex Palytopes Walfgmg spiegel

67

Convexity Theorems for Generalized Planar Configurations Jacob E. Goodmm and Richard Pollack

73

Is There a Krasnoselskii Theorem for Finitely Starlike Sets?

Bruce E. Peterson

81

Convex caustics for Billiards in R2 and R3 Philip H. Turner

55

0n Packing Curves into Circles

Emir; Luwak A Perspective on Abstract Conveirity:

107 Classifying Alignments by

Varieties

Robert E. Jamison-Walther

113

CONTENTS

Vi Open Problems Around Radon'a Theorem

John R. 1?e Generalizations of Kelly‘s Theorem; Open Problems Gerard sierksma

151 173

Unimorphiea of Subsets of Hausdorff Locally Convex Vector Spaces

René Foumeau

193

Tiling Rd by Translates of the Otthants

Jim Laurence Eberhard'a Theorem for Convex 3—Polytopea Joseph Malkemltch

203 209

Graphical Difference Sets and Projective Planes

Andrew Sobcayk

215

Tiling the Plane with Incongruent Regular Polygons

Hans He

225

Problems

229

Author Index

235

Problem Index

237

Subject Index

239

CONTRIBUTORS

David Barnetce, Department of Mathematics, University of California at Davis, Davis, California.

Arne hindered, Institute of Mathematics, University of Copenhagen, Copenhagen, Denmark. G. D. Chakerian, Depm-tment of Mathematics, University of California at Davis, Davis, California. 'René Fourneau. Unité de Mathématiq'ues, Institut Superieur Industriel Diégeois, Liége, Belgium.

Paul Goodall,"c Department of Mathematics, University of Oklahoma, Norman,

Oklahoma. Jacob E. Goodman, Department of Mathematics, The City College, City University of New York, New York, New York. Hans Herdafi Department of Applied Mathemtics, the Weizmann Institute of Science, Rehovat, Israel. Robert E. Jamison-Waldner, Department of Mathematical Sciences, Clemson University, Clemson, South Carolina. Victor Klee, Department of Mathemtics, University of Washington, Seattle, Washington. Jim Lawrence, Department of Mathematics, University of Kentucky, Lexington, Kentucky.

Carl Leaf Center for Applied Mathematics, Cornell University, Ithaca, New York.

Current affiliation: Department of Mathematics, Royal Holloway College, landon University, London, England. +Current affiliation; ton, Massachusetta.

Department of Mathematics, Boston State College, Bos’

*Current affiliation: Department of Mathematical Sciences, IBM T. J. Watson Research Center, Yorktown Heights, New York; and Deparbnent of Mathematical Sciences, University of Kentucky, Lexington, Kentucky.

v‘H

v1' 1' 1'

CONTRIBUTORS

Erwin Lutwak, Department of Mathanatica, Polytechnic Institute of New York, Brooklyn, New York. Joseph Malkevitch, Department of Mothemtiaa, York College, Jamaim, New York. Bruce E. Peterson, Depca-tment of Mathematica, Middlebury College, Middlebury, Vermont. Richard Pollack, Deparhnent of Mathematics, (7t Institute of Mathematics, New York University, New York, New York. John R. Reay, Department of Mathematics mud Computer Science, Western Washington University, Bellinglkm, Washington. G. Thomas Sallee, Depm‘tment of Mathematics, University. of California at Davis, Davis, California. Gerald Sierksma, Mdepartment of Mathematics, Econometric Institute, lini— oersity of Groningen, Grouinyon, The Netherlands. Andrew Sobczyk, DepaI'brIent of Mathematical écienaes, Clanson University, Clemson, South Carolina.

Wolfgang Spiegel, ” ” 'oh 7 l" ‘1' ‘ik), Vuppertal, Federal Republic of Gemany.

1

1‘

L 7‘

v-rr

1:

Philip H. Turner,* Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana.

1——

.

.

.

Current affiliation: Department of Systems Fareoaet, United Illuminating Company, New Haven, Connecticut.

HOW MANY STEPS? Victor Klee Department of Mafltematics ~Un£uersity of Washington Seattle, Washington

Stimulated since the early 1950's by its relationship to linear programing and more recently by other connections with computational questions,

the

combinatorial study of convex polytopes has advanced greatly in the last 25 years.

For an overview the reader may skim successively the survey ar-

ticles or books of Klee 1966 [14], Grunbaum 1967 [8], Grfinbaum and Shepherd 1969 [11], Grlmbaum 1970 [9], HcHullen and Shepherd 1971 [21], Grunbaum 1975 [10], Klee 1975 [15], and Barnette 1978 [3] — and then read the recent proofs by Billera and Lee [lo] and Stanley [23] of an important 1971 conjec— ture of Mcflullen [20]. The d—step conjecture was first formulated by w. M. Hirsch in 1957 (see Dantzig [5,6]), and despite progress on many other fronts it remains unsettled.

Because of intrinsic interest and connections with questions of com-

putational complexity, and because solutions may well require the development V of deep new methods, the d—step conjecture and its relatives are probably the most important open problems on the combinatorial structure of convex polytopes.

The present-note, which is extracted from a longer article in prepa-

ration, states several equivalent forms of the d-step conjecture.

Some deal

explicitly with edge—paths on pnlytnpes, one involves matrix pivot operations, and one concerns an exchange process for simplieisl bases.

1

2

KLEE Suppose that X is the nonnegative orthsnt R: in real d-space Rd, x is

the point (0.0....,0), x' is the point (1,1,...,l), and X' = x' — late of the nonpositive orthant 41:.

i, a trans-

Then

(a)

X and x' are affine orthants with respective vertices x 6 int X' and x' E int X , and

(b)

the intersection P - X l'] x' is bounded.

One form of the d-step conjecture asserts that whenever conditions (a) and (b) are satisfied the uertices x and x' of the polytape P can be joined by

a path formed from at most d edges of P.

That is certainly true in the exv

ample, for there 1’ is the unit cube [0,l]d.

However, even when (a) and (h)

are augmented by the requirement that P is simple (each vertex incident to

precisely d edges) there are many other possibilities for P. In particular, the possible number of vertices then ranges from cl2 - d + 2 to Mat]; 1)

when d = 2k and to H3"; 1) when d = 2k + 1. The d-polytapes of the above form X n X' have precisely 2d facets (faces of dimension d - 1).

Hence a formal strengthening of the above conjecture is

the conjecture that A(d,2d) - d, where A(d,n) denotes the maximum diameter of d-polytopes P with n facets.

(That is, A(d,n) is the smallest integer k such

that any two vertices x and x' of such a P can be joined by a path formed from at most It edges of P.)

A(d,n) S n - d.

A further formal strengthening is the conjecture that

However. it was proved by Klee and Walkup [17] that these

conjectures are all equivalent, though not necessarily on a dimension—fordimension basis.

Another equivalent conjecture is that any two vertices of

a simple palytape can be joimd by a path that does not revisit any facet

[17]. Turning now to matrix pivot operations, we note that the d-step conjecture is equivalent to the following:

If the real d x (2d +1) matrices A = (I,B,c) and A' = (B',I,c') are rowequivalent, where d z 2, I is the d x d identity matrix, and the columns c

and c' are > 0 , and if the polyhedron P = {x 6 Kid : (I,B)x = c)

is bounded, then it is possible to pass from A to A_' by a sequence of S d feasible pivots followed if necessary by a pemtatian of rows.

’

HOH MANY STEPS?

3

Here a pivot, as applied to an m X (n + 1) matrix S - (s1 ), is the operation

of choosing (1,1) with j E n and Bi] 1‘ 0, then dividing the i—th row of S by sij so as to obtainll in position (1,1), and finally subtracting appropriate multiples of the i-th row from other rows so as to obtain 0 in all positions (h,j) for h 1‘ i.

A pivot is feasible if the last column of the

matrix is nonnegative both before and after the pivot.

of the several forms of the d-step conjecture presented here. the present version is closest to Hirsch's original'form [5, pp. 160 and 168] and is most closely related to linear programing methods.

In the example below,

d - 2 and the pairs (1,1) under the arrows indicate the positions of the pivot -tries .

l

0

0

Z

-l

2

1/2

0

1

-1/2

1

(1,3)

1

—1

2

(I

B

2

>

2/3

1/3

1

1/3

2/3

0

D

2

(2.4) 1/2

1

O

3/2

3

c)

(B'

l

I

2

c')

A set B C ltd-1 is a simpliciaZ basis (also called a minimum positive basis) for

d"' if it is the vertex-set of a (d — 1)-simp1ex whose interior

includes the origin.

Equivalently, B is affinely independent, is of cardi-

nality d, and the origin 0 is a strictly positive combination of the points of 3.

Another equivalent form of the d-step conjecture is reminiscent of the

exchange process used to show 311 linear bases of a vector space are of the same cardinality.

It is as follows:

If B and 81 are disjoint simplicial bases of Rd—1 and the union U = B U 131 is a Gsar set (every set of d — 1 points of U is linearly independent), then

there is a sequence B - BO’BI""’Nd - B1 of simplicial bases such that for I. 5 i E d, Bi is obtained from 31—1 by replacing a point of 51-1 with a point

of U N 31-1' In the example below, d - 3 and 0 < s < 1.

1

o

0

1

-1

-1

(so = B)

———->

.

1

o

0

1

-5

-1

(31)

The rows represent points of Bi'

1 1—:

. 1 1—5 —>

p

0

1

-s

-1

(32)

——>

—1

-s

-s

-1

= , (53 B )

4

KLEE As evidence in favor of the d—step conjecture. one might cansider the

facts that it is obvious when (1 € {2,3}, is easily proved when d = A [13], and has been proved for d = 5 [15].

In fact, the result for d = 5 has been

extended by Adler and Dantzig [l] to a much more general class of combinatorial structures.

the conjecture has been proved,

Also,

for arbitrary d,

for polytopes arising from certain sorts of linear programs (see [15] for references, and see especially Provan and Billera [22]). As evidence against the d-step conjecture, we note that when stated without the boundedness condition (1:), it is correct when d E {2,3} but not when d = 4 [17].

other strengthened forms of the conjecture have been dis—

proved by Walkup [25], Mani and Walkup [19], and Todd [2A]. It seems likely that the conjecture is false for d = 12 and perhaps even for d = 6.

If that is so, what can be said about the asymptotic be—

havior of A(d,2d) as d -> .2 known counterexamples

[19,

Does A(d,2d) increase linearly with d'! 24,

seem to be only "linearly bad.") tially’.l

(The

25] to strengthened forms of the conjecture

Quadratically‘!

Polynomially?

Any of these conclusions would be of great interest.

ExponenIf it could

be shown that A(d,Zd) is bounded above-by a polynomial in d, the resulting insight might lead to a new pivot rule for the simplex method of linear pro— graming which would carbine the practical advantages of Dantzig's pivot rule with the theoretical advantages of the Shor—Khachian algorithm.

‘

(Dant—

zig's algorithm is excellent in the practical sense [5, 7], but its worst— case behavior is exponentially bad [16].

The Shor—Khachian algorithm is

"good" in the sense of being polynomially bounded [12], but is not a useful practical tool in its present form [7].)

If it could be shown that

A(d,2d) increases exponentially with d, that would indicate a strong limitation on the worst-case efficiency of any edge-following algorithm for linear programing.

Though sharper results are known for a few small values of d and n - d (see [15] for references), the best general bounds on A(d,n) are the follow-

ing, due respectively to Adler [1] and Larman [18]: [(n

I

_

d)

_ {n - d) [id/1+]

+ 1 S A(d,n) S 2 d-3 n

n particular, d s A(d,2d) S 2 d-3 d.

How MANV STEPS?

5

REFERENCES I. Adler. Lower bounds for maximum diameters of polytopes, Math. Pro: gzwrming Study 1(1974), 11—19.

I. Adler and G. B. Dantzig.

Maximum diameter of abstract polytopes,

’ Math. Pragmming Study 1(1974), 204.0. D. W. Barnette. Path problems and extremal problems for convex polytopes. Relations between Comimtorica and other Parts of Mathematics (D. K. Bay chaudhuri, ed.), Amer. Math. Soc. Proc. Symp. Pure Math. 34(1979). 25-34. ' L. Billera and C. Lee. Sufficiency of mnullen's conditions for fvectors of aimplicial polytopes, Bull. Amer. Math. Soc. 2(1980), 181185.

G. B. Dantzig. Linen Progrmrring and Extensions. sity Press, Princeton, N. J., 1963.

Princeton Univer—

G. B. Dsntzig. Eight unsolved problems from mathematical programing, Bull. Amer. Math. Soc. 70(1964), 1099-500. G. B. Dantzig. Comments on Khachian's algorithm for linear programing, Tech. Report SOL 79—22, Dept. of Operations Research, Stanford University, 1979. 11. Grunbaum. Convex Polytopas. science, New York, 1967.

Pure and Appl. 143:1... Vol. 16, Inter-

B. Grlinbaum. Polytopes, graphs and complexes, Bull. Amer. Math. Soc. 76(1970), 1131-1201. 10.

B. Grunbaum. Polytopal graphs, Studies in Graph Theory, Part II (D. R. Fulkerson, ed.). Math. Assoc. Amer. Studies in Math. 12(1975), 101—225.

11.

B. Grflnbaum and G. C. Shepherd. Soc. 1(1969), 257-300.

12.

1.. G. Khschian. A polynomial algorithm in linear programing, Soviet Math. Daklady 20(1979), 191-195. (Translated from Dokl. Akad. Nauk SSSR 244(1979), 1093-1096.)

13.

V. Klee. 602-614.

14.

V. Klee. Convex polytopes and linear programing, Proc. IBM Sci. Com— put. Sympos. Combinatorial Problems, Yorktown Heights, N. Y., 1964, IBM

15.

V. Klee. Convex polyhedra and mathematical programing, Pt'ac. International Congreas of Mzthmticiane, Vancouver, Canada, 1974, Canadian Hath. Congress, 1975, 485-490.

16.

V. Klee and G. J. Minty. How good is the simplex algorithm? Inequalities III (0. Shiaha, ed.), Academic Press, N. Y., 1972. 159-175.

17.

V. Klee and D. W. Walkup. The d-‘step conjecture for polyhedra of dimension d < 6, Act: Math. 117(1967), 53-78.

18.

D. G. Larman. 161-178.

Convex polytopes, Bull. London Math.

Diameters of polyhedral graphs, Canad. J. Math. 16(1964),

Data Process Division, White Plains, N. Y.,

1966,

123-158.

Paths on polytopes, Free. London Math. Soc. (3)20(1970), ' *

KLEE 19.

P.Man1 and D. W. Halkup.

20.

1’. McMullen.

A 3-sphere counterexample to the Wv-path

conjecture, Math. of aperat'tlone Res. 5(1980), 595—598. The numbers of faces of simpliciul polytupes, Israel J.

Math. 9(1971). 559-570. 21.

P. mmdlen and G. c. Shepherd. Convex Palytopss and the Upper Bound Conjecture. London Math. Soc. Lecture Note Series, 3, Cambridge University Press, Iondou,

1971.

22.

J. S. Proven and L. J. Billera. Decompositiona of simplicial couplexes related to diameters of convex palyhedra, Math. of Operations Res. 5 (1980). 576-594.

23.

R. Stanley. The number of faces of e simplicial convex palytope, Advances in Math. 35(1980), 236-238.

24.

M. .1. Todd. The monotonic bounded Hirsch conjecture is false for r11mensicn at least A, Math. of Operations Res. 5(1930), 599-601.

25.

D. W. Walkup. The Hirsch conjecture fails for triangulated 27-spheres, Math. of Operations Rea. 3(1978), 224- 230.

POLVHEDRAL MAPS 0N Z-MANIFOLDS David Barnette Department of Mathematics University of California at Davis Davis, Califmia

1.

INTRODUCTION

Graphs embedded in 2—dimensionsl manifolds have been studied for about 1.00 years.

Properties involving embeddings and colorings of graphs have been

extensively studied.

There are, however, avnumher of problems similar to

ones considered for convex polyhedrs which have not been investigated.

For

example, a theorem of Steinitz [8] tells us that any map on the sphere which is 3-connected is isomorphic to the graph of some convex 3-dimensional polytope (herufter to be called 3-palybapss).

If a map is drawn on some other

2—dimensionsl manifold,» very little is known shout when there exists a polyhedron—like structure isomorphic to it.

Questions like this will be consid-

ered here and properties of such polyhedron-like structures will be examined.

2.

POLYHEDRAL MANIFOLDS

Most of the structures to be considered are examples of 2-ce11 complexes.

A

Z-csZZ coupler is s collection of convex polygons such that the intersection of any two is a vertex of both, an edge of both, or is empty.

The polygons

- will be called the faces (or sometimes the facets) of the complex.

8

BARNETTE A toroidal polytape is a Z-cell complex whose union is a torus such that

no two faces meeting on an edge are coplanar.

A polyhedral 2—manifold is a

Z—cell complex whose union is a 2-manifold (embedded in some Euclidean space)

such that no two faces meeting on an edge are coplanar. Two examples of toroidal polytopes are the triangular picture frame (Fig.

l) and the famous Csdszfir Polyhedron which is a toroidal polytope with seven FIGURE 1.

—/

vertices, each two joined by an edge.

(For instructions on how to build one,

see Excursions into Mathemtics by Beck, Bleicher, and Crow, Chapter 1.) We shall use the standard method of representing toroidal maps as maps drawn on a rectangle with the top and bottom to be identified and also the two sides identified.

The graphs of these two examples are shown in Figs.

‘4... A... -.J- .__’._A_

FIGURE 2.

_.A

2 and 3, respectively.

POLYHEDRAL MAPS 0N Z-MANIFOLDS

9

FIGURE 3.

The 3-connectivity of maps on the sphere is equivalent to the property that no two faces have a multiply connected union.

For maps on the sphere

this is sufficient to guarantee a corresponding polytope. 'we shall see that this is not sufficient.

For toroidal maps

It is clearly a necessary condi-

tion because two‘convex polygons cannot have a multiply connected union.

We

shall say that a map on a 2—manifold is polyhedral provided each face of the map is a 2-ce11 and no two faces have a multiply connected union. It will be useful to be able to talk about polyhedra that have self intersections, so we shall distinguish between an embedding and an immersion of our polyhedron in some Euclidean space.

An embedding of a polyhedral map

M is a homeomorphic image of M in E“ such that the image of each face of the map is a convex polygon and no two polygons meeting on an edge are coplanar. Embeddings will also be called realizations of the map.

An immersion of a

polyhedral map is a continuous image of M in E1‘ such that each face of M is taken one—to—one onto a convex polygon and no two polygons meeting on an edge are coplanar.

In other words, an embedding is a polyhedral manifold, while

an immersion is essentially a polyhedral manifold with self intersections. To see that polyhedrality is not a sufficient condition to guarantee that a toroidal map be realizable, a very large family of such maps will be proved to have no immersions in any Euclidean space.

THEOREM.

There are no polyhedral immersions of 3-valent polyhedral maps on

orientable manifolds of genus g > 0.

Proof.

Let s be the sum of the two—dimensional angles of the faces at

their vertices.

We use a normalized angle measure in which a 360° angle has

BARNETTE

10 a measure of l.

If we

The sum of the angles of an n—gon is thus (n - 2)/2.

let p1 be the number of i-sided faces of any such immersion, then s -

2(1 — 12)!)1/2 which equals —Ei.pi — )Zp1 which is just )3 — F, where E and F are the numbers of edges and faces, respectively.

Another way to get a is to Since each ver—

add the angles around each vertex, doing so for uch vertex.

tex is 3-valent, the sum of the angles at the vertex is less than 1, and s < V, where V is the number of vertices.

We now have that V > E - F.

Combining

this with Euler's equation, V - E + F = 2 — 23, we get g < 1, thus, g - 0. This argument clearly holds for all nonorientable 2-manifolds except the projective plane.

The theorem is also true for the projective plane but a

different proof seems to be necessary. We now turn to an interesting example to be called the twisted triangflAZar picture firms (Fig. 4).

It resembles the triangular picture frame,

FIGURE 1».

but upon examining it one will see that it does indeed have a twist in it. when the author's thesis student, Jean Simutis, was working on problems

of realizing toroidal maps as polyhedra, he drew her a picture of a twisted triangular picture frame and asked her to give a geometric construction of one.

The author felt at the time that from it and the triangular picture

frame one could construct a great number of the possible toroidal polytopes.

Simutis was very slow to provide the required construction.

Even after the

author made suggestions on how to begin she couldn't seem to carry it through. He became rather impatient with her. to construct it!

It is almost clear from the picture how

We finally discovered why she was having so much trouble

when she proved that it didn't exist (as a toroidal polytope).

Another proof

(in fact two proofs) were done independently by a student at Stanford University named Scott Kim.

Kim also proved that the Io-sided twisted triangular

picture frame and the lo-sided quadrilateral picture frame are realizable

(Figs. 5 and 6).

‘

POLYHEDRAL MAPS 0N Z-MANIFOLDS FIGURE 5.

11 FIGURE 6.

/

‘0'

There are very few general teslizability theorem for toroidal maps.

One by Altshuler [1] deals with triangulations.

' THEOREM.

If T is a triangulation of the turns and if T contains a simple

circuit C such that each triangle has an edge on C,

then T is realizable as

a toroidal polytape.

There is also a theorem of Simutis [7] which gives us some idea of which maps are the hardest to show are realizable.

We shall let Q be the set of all

maps on the torus such that all faces are h—sided and all vertices are A—vslent.

THEOREM

(Simutis).

et splitting“ and if

If M and M' are not in Q. if M' is obtained from H by fac— M

is realizable, then so it 14'.

In her work, Simutis found a striking similarity between a well-known theorem for 3-polytopes and a property of toroidal polytopes.

Using Euler's

equation and a few simple observations and constructions one can show that

the minimum number of edges for a 3-polytope is 6, there is no 3—polytope with exactly 7 edges, but for all n > 7 there exists a 3—polytope with ex— actly n edges.

Simutis showed that the minimum number of edges of a toroidal

polytope is 18 (the triangular picture frame), there is no toroidal polytope with exactly 19 edges but for all n > 19 there is a toroidal polytope with ex-

actly u edges .

*Faca't splitting is a process of adding an edge across a face to obtain a new map with one more face than before.

12

BARNETTE We shall now list some open questions dealing with realizability of poly—

hedral maps .

1.

Are there any other neighborly polyhedra besides the tetrahedron and the Csaszar polyhedron? (The mdifier neighbarly means that each two vertices are joined by an edge.) Easy calculations using Euler's equation show that the simplest unsolved case is the triangulation of the otientable surface of genus 6, having 12 vertices. What is "Steinitz's" theorem for tori?

(i.e., what are necessary and suf-

ficient conditions for a toroidal map to be realizable?)

Which twisted picture frames are realizable? For each m and n greater than 3 there is an m—sided n-gonal picture frame. Furthermore, there are [n/Z] different types of twists that the picture frame can be given. (Conway).

Is there a toroidal polytope that can only be realized in a

knotted fashion? Conway posed this question about 15 years ago and stated that he believed that we were almost to the point where we could answer such a question. Are the triangulations of the torus all realizable in ES? We list this as unsolved although Peter Mani claims to have proved that they are all realizable. He made this claim seven or eight years ago and a written version is yet to be seen.

Are the triangulations of the other orientable manifolds all realizable in Es? Are the triangulations of all 2—msnifolds realizable in El"!

Since every

simplicial 2-complex is a subcomplex of a Schlegel diagram of some 6—

polytope, it follows that they are all realizable in E5.

For the ori—

entable manifolds the question is whether they can be realized in 123 or E .

Since the nonorientable manifolds are not topologically realizable

in E3, the only hope is for realizations in E .

'

what about manifolds of dimensions greater than 2’! This is a rather vague question because part of the problem is to determine what questions should be asked about them.

with regard to Question 7, it can be shown that the triangulations of 4 [4]. The idea is to decompose

the projective plane are realizable in E

the triangulation into a Schlegel diagram of a 3-polytope and a mbiue

strip, then show that the Hobius strip can be realized in E“ with its ver-

tices arbitrarily close to the corresponding vertices of a polytope iso— morphic to the Schelgel diagram.

Once this is done it is easy to nave the

vertices of the polytope to effect a gluing of the two pieces, producing

the desired polyhedral manifold in E4.

'

POLYHEDRAL MAPS 0N Z-MANIFOLDS 3.

13

CONVEX AND NONCONVEX VERTICES

In this section we shall deal only with orientable manifolds.

We shall say

that a vertex of a polyhedral manifold in E3 has :1 vertex figure provided there is a plane that separates the vertex from its neighbors (i.e., from the

vertices to which it is joined by edges).

If a vertex has a vertex figure we

define the vertex film to be the intersection of such a separating plane with the faces that meet the vertex.

A vertex is convex provided it has a

_convex vertex figure. It is clear that a convex 3-polytope has-only convex vertices.

It is al—

most clear that polyhedral manifolds of higher genus must have nonconvex ver— tices.

This can be seen by observing that if all vertices are convex then the

sum of the angles of the faces at a vertex is less than 1.

The argument that

showed that there are no 3-va1ent imersions now can be carried through to show that there are no polyhedral manifolds with all vertices convex for genus _ greater than 0.

(Since we are now treating realizations in E3, the projective

plane, which is a case not covered in the argument,

does not have to be con—

sidered.) Another way to see that there must be nonconvex vertices is to observe that orientable surfaces of genus greater than 0 must have saddle points with respect to some direction. tices. points.

Saddle points can be shown to be nonconvex ver-

It would seem that for a torus there should be at least two saddle (Stick a pencil through the hole.

Move it up as far as you can and

the notion of the pencil should eventually he obstructed by a saddle point. Similarly a saddle point should be found by moving the pencil downward.)

It

also seems that as the genus increases, the number of saddle points should increase; after all, there are more holes to stick your pencil through. All this is true if you are dealing with smoth manifolds, but with poly-

hedral manifolds strange things happen.

Banchoff [2] has constructed a

toroidal polytope with only one saddle point in one particular direction (in other directions there are more saddle points).

Even more incredibly,

he constructed orientable polyhedral manifolds for every positive genus,

each with only one saddle point in one particular direction.

This doesn't settle the question "of how many nonconvex vertices polyhedral manifolds must have.

It shows that searching for saddle points probably isn't

the way to go about‘it.

14

BARNETTE Another problem the author gave to Simutis was to prove that every to-

roidal polytope has at least six nonconvex vertices.

Just as she did with

The

the twisted triangular picture frame, she had difficulty proving it.

author provided her with a heuristic argument: has an entrance and an exit.

A torus has a hole.

A hole

Around an entrance one should get at least three

nonconvex vertices and similarly, three more around the exit.

She seemed to have difficulty making this argument work and finally aettled for proving that there must be at least three nonconvex vertices in a toroidal polytope. The author has since proved that there are at least four nonconvex ver-

tices and has constructed a torus with nine vertices that has only five non— convex vertices (see [3]). The author has also constructed polyhedral manifolds of all genuses with The construction is quite 31m-

exactly seven nonconvex vertices (see [3]).

lar to Banchoff's. The edges of polyhedral manifolds in E3 are of two types.

We shall say

that an edge is concave provided a small disc lying inside the manifold locally supports the edge (note that the manifold separates E3 into an inside and an outside region).

An edge that is not concave is called convex.

A

nonconvex vertex of an embedding of an orientable polyhedral manifold will

have both convex and concave edges meeting it.

This is not true, however,

for immersions. Since every toroidal polytope has nonconvex vertices, it is clear that every toroidal polytope has concave edges.

In fact every one must have at

least two concave edges because there are at least four nonconvex vertices. 0n the other hand there is an embedding of the triangular picture frame with only three concave edges.

It is not known if every toroidal polytope has

at least three concave edges. We conjecture that every toroidal polytope has at least three concave edges and that as the genus of an orientable polyhedral manifold embedded in E

increases, the minimum number of concave edges it can have will also

increase.

For iumnersions, however, the author conjectures that there is a

bound on the

number of

edges

of the genus.

Here are some more unsolved problems:

10.

Must all vertices of any realization of the Csaszér polyhedron be nonconvex?

Are there realizations of the Csdsrar polyhedron that are "different" from the usual one?

heme . 1

9.

POL‘IHEDRAL MAPS 0N Z-MANIFOLDS

15

11.

What do we mean by "different realizations" of two polyhedral manifolds?

12.

Are there realizable maps on the torus that have no "inside out" realization? An example of what is meant by an inside out realization is the four-sided triangular picture frame which is an inside out realization of the three-sided quadrilateral picture frame.

13.‘

Does there exist a combinatorial type of polyhedral manifold such that every realization must have a convex vertex of valence Z 4 7

11».

What goes on in higher dimensions? How would we classify the convex nature of faces of various dimensions? How many of the various types must occur in the various types of manifolds?

’15.

Is there some kind of relation (perhaps Euler type) between the number of convex vertices,

4.

nonconvex vertices, ‘concave edges,

etc.’!

GENERATING MAPS AND TRIANGULATIONS A theorem of Steinitz [8] implies that the triangulations of the two—

sphere can be generated from the boundary of the tetrahedron by a process called vertex splitting. cident edges are chosen.

In vertex splitting, a vertex and two of its inThe vertex is replaced by two vertices and the

two edges are replaced by two triangles as in Fig. 7. FIGURE 7.

A number of years ago Griinbaum and Duke worked on generating the tri— angulations of the torus.

They found a set of 22 triangulations of the torus

from which one should be able to generate all others by applying sequences of vertex splittings. The inverse of vertex splitting is called edge shrinking.

The 22 tri—

angulations of Griinbaum and Duke were thus minimal with respect to edge

shrinking.

The generating problem for triangulations of 2—manifolds is

really the problem of finding the minimal triangulations.

What GrUnbaum and Duke wished to do was to use these minimal triangulations to show that all triangulations of the torus were realizable as toroidal polytopes. realizable,

They planned to show that all minimal triangulations are

and that realizations of the other triangulations could be

16

BARNETTE They

constructed from them by doing a kind of "geometric" vertex splitting.

were never able to show how the "geometric" vertex splittings could be done and they finally abandoned the project.

The minimal triangulations remained

ignored in Gri‘mbaum's files until a few months ago when this author asked his thesis student, Kurt Rusnak, to see if he could give a proof that this was the complete set of minimal triangulation (Grflnbaum anti Duke never wrote a proof). He soon found two more minimal triangulations and he seems to have proved that he has found all of them.

THEOREM

Thus, we have:

(Grunbsum, Duke, Rusnsk).

The triangulations of the torus can be

generated from a set of 24 minimal triangulations by vertex splitting. The generating problem has been solved for one other manifold.

’

The su-

thor has recently shown [5] that the triengulations of the projective plans can be generated from the following triangulations (Fig. 8).

FIGURE 8.

l

FIGURE 9.

%_§

1

~94

%

POLYHEDRAL MAPS 0N Z-MANIFOLDS

17

Onevcan also consider generating all polyhedral maps of a manifold.

By

using a more general family of vertex splittings (see Fig. 9), one can generate all polyhedral maps on the 2-sphere starting with the boundary of the tetrahedron [8].

For ‘the projective plane and the torus it appears that the

set 'of minimal maps will be very large.

However, if we admit two processes,

vertex splitting and its dual, facet splitting, then the set of minimal maps is much smaller.

For the projective plane we believe that the following is

the set of minimal maps (171g. 10).

I

FIGURE 10.

We have the following unsolved problems on generating maps: 16.

How does one generate the polyhedral maps on the torus!

17.

How does one generate triangulation or polyhedral maps on manifolds of higher genus?

18.

For every type of 2-manifold is the set of minimal triangulations and

19.

How does one generate other types of maps such as 2-cell embeddings?

the set of minimal maps with respect to edge shrinking finite?

18

5.

BARNETTE

A NEW FOUR-COLOR CONJECTURE Before the four-color theorem was proved, it was well—known that the

four-color conjecture was true if and only if it was true for convex 3— polytopes.

It was also well—known that maps on the torus are 7—colornble

and that some maps require seven color].

It is a little surprising that

toroidal polytopes are G-colorable.

One can prove the fi-coloring property by showing that in the dual graph one can successfully remove vertices of valence at most five until there are at most six vertices left, then coloring the six vertices and returning the vertices one at a time choosing a color for each as it is returned.

The au-

thor has shown that if there is a polytope for which this is impossible then

there exists an imersion of a polyhedral map with all vertices 3-va1ent [6]. We have not been able to find a toroidal polytope that actually requires six colors.

In fact, we cannot find one that requires five colors. and we

conjecture that all are 4-colorable. color conjecture.

This, however, is not the author's A-

We conjecture that all polyhedral manifolds are A—color—

able. It hardly seems that this conjecture could be true, yet we can find no counterexample.

One difficulty is that we cannot find any polyhedral mani-

folds for which all faces have a large number of edges. We conclude with two last questions: 20.

Does every polyhedral manifold have a face with five or fewer edges?

21.

Does there exist any n such that every polyhedral manifold must have a face with n or fewer edges?

REFERENCES

l.

A. Altshuler. Polyhedral realizations in 33 of triangulations of the torus and 2-manifolds in cyclic 4-polytopes, Discrete Math. 1(1971), 211-238.

2.

T. F. Banchoff. Critical points and curvature for embedded polyhedral surfaces, Am. Math. Monthly 77(1970), INS-ASS.

3.

D. Barnette. Nonconvex vertices of toroidal polytapes, to be published in Israel J. Math.

1».

D. Barnette. unpub 1 ished .

Realizing triangulatious of the projective plane in 1:“,

5.

D. Barnette. lished.

Generating triangulations of the projective plane, unpub-

POLYHEDRAL- MAPS 0N Z-MNIFOLDS

19

6.

D. Barnette.

Coloring polyhedral manifolds, unpublished.

7.

J. Simutis. Geometric realizations of toroidal maps, Ph.D. Thesis. University of California, Davis, 1977.

8.

E, Steinitz and E. Rademacher. Berlin, Springer—Verlag, 1934.

Vorlesungen fiber die thear-ie tier palyeder,

CHARACTERIZING THE NUMBERS 0F FACES OF A SIMPLICIAL CONVEX POLVTOPE Carl W. Lee

center- for Applied Mathematics Cornell University Ithaca, New York

1.

INTRODUCTION

Polyhedra have long been an object of study, but in the last few decades both the development of linear programing and the expansion of the field of combinatorics have awakened deep interest in d—dimensionsl polyhedral.

0f the

many combinatorial questions one may ask about polyhedra, we will single out one:

how many faces of various dimensions can a polytope, particularly a sim-

plicial polytope, have? We will recount highlights of endeavors to answer this question. while remaining somewhat faithful to historical chronology. A convex polyhedron is the intersection of a finite number of closed half-spaces in Rd. sion d a 1.

By a d—polyhedr'an we mean a convex polyhedron of dimen—

A convex polytope is a bounded convex polyhedron; equivalently,

it is the convex hull of a finite number of points in Rd.

For d-polyhedron

1’ let Ej(P) I Ej denote the number _of j—dimensional faces of P,

We shall use the convention f_1(P) = 1 throughout this paper. f(P) = (fo(P),...,fd_1(P)) is the f-vector of P.

O S j S d — 1.

The d-vector f =

Faces of P of dimension 0, 1

and d — l we will call vertioes, edges and facets, respectively.

A polytope P

is simplicial if every_ face of P is a simplex; i.e., if every j-dimensional face of P contains exactly j + 1 vertices of P.

Gtiinbaum [18] may be used for

'*L‘urrent affiliation: Department of Mathematical Sciences, IBM T.J. Watson Research Center , Yorktown Heights, New York; and Department of Mathematical Sciences, University of Kentucky, Lexington, Kentucky

21

22

LEE

referred to for details of notions in the theory of polytopea and major re-

sults in this field through 1967. Let Pd be the class of all d—polytopes and let P: be the class of all simplicial d-polytopes, and define

£(Pd) = {5(a) : y e Pd}

:1 _— my) .. P 6 PS} d £023) We will concentrate primarily upon f(P:) .

The recent complete charac-

terization of this set is a major step toward the problem of describing the

f—vectors of all polyhedrs, and more generally toward the even more chal— lenging task of the classification of the combinatorial types of polyhedra.

There are several advantages to focusing attention on simplicial polytopes.

They are a "natural" class in the sense that "in general" no more

than d + 1 points chosen from R

lie on a common hyperplane.

Secondly, the

class of simplicial d—polytopes is dual to the class of ainple d-polytopea ——those with the property that every vertex is on precisely d facets—«relevant to the theory of linear programing. polytopes,

Thirdly, many problems about arbitrary

such as the Upper Bound Theorem, are reducible to ones concerning

simplicial polytopes.

Finally, the face lattice of a simplicisl polytope can

be examined within the more general context of simplicial complexes, allowing

the possibility of the application of algebraic techniques [18, 54.5].

2.

EULER'S RELATION

Euler's discovery in 1752 of the relation to - fl + £2 = 2 for 3-polytopes [16, 17] has been acclaimed as "the first important event in topology"

(Alexandroff—Hopf [1, p.11) and as "the first landmark" in the theory of polytopes (Klee [21]).

The following theorem provides the generaliution

of this relation to higher dimensions.

THEOREM 1 (Euler's Relation). If f is in 50’“) then d—l

E «nitj - 1 + (.1)“'1

CHARACTERIZING NUMBERS 0F FACES OF A POLYTOPE

23

Schlifli [38] formulated Euler's Relation for d > 3 in 1852 and Poincare [36] in 1399 provided the first real proof of this generalization.

Grfinbaum

[18, 58.2] introduced in 1967 a completely elementary, nonalgebrsic demonstra— tion of Theorem 1.

In [18, §8.6] Gtfinbaum sketches the history of investiga-

tions into Euler's Relation and explains that research into its applicability and extensions helped direct attention to the idea of convexity.

Imre Lakatos

[26] chooses the interesting evolution of Euler's Relation to illustrate his philosophy of the process of mathematical discovery. It is instructive to observe that quite a few of the early attempts to prove the theorem assumed that the boundary complex of a d—polytope P is

shallabls, that the facets of P can be ordered F1,F2,...,Fm so that Pk n (UlsiSk-lFi) is homeomorphic to a (d — 2)-ball, 2 S k S m - 1.

This asser-

tion was not established, however, until 1971 by Bruggesser and Mani [13] and has proven to be a powerful tool in the study of polytopes.

3.

THE DEHN-SOMMERVXLLE EQUATIONS

In terms of linear relations satisfied by all f-vectors of d—polytopea, Euler's Relation is the best possible.

Restriction to the class of simplicial poly—

topes, on the other hand, allows a significant strengthening of Theorem 1, offering the first evidence that f(P:) is a more tractable set than f(?d).

THEOREM 2 (The Dehn-Sommerville Equations).

a-1 a 12k (sk) (-1) 3 (kj + + 1I):j _- (-1) d—l ER,

If f is in f(P:) then

—1 s k s a - 2

It is easy to see that (Ed _1) is Euler' sdRelation and it has been shown that exactly [(d + 1)/2] of the equations (Bk) are independent. where [-1 denotes the greatest integer function. In 1905. Dehn [15] worked on the relations for d = 4 and d = 5 and conjectured the existence of analogous relations for d > 5. derived the complete system for arbitrary d in 1927.

Sommerville [39]

Klee [20] in 1964 re—

discovered the Dehn-Sommerville equations in the nmre general setting of manifolds and incidence systems. ‘In addition to boundary complexes of sinr

plicial polytopes, Theorem 2 applies also to triangulations of topological and homology (d - 1)-spheres and Klee's Eulerian (d - 1)—spheres.

'Ch. 9] for more historical details and generalizations.

See [18,

24

LEE The next stage in the development of the Dehn-Sonmetville equations was

their recesting in an especially useful manner by HCHullen and Walkup [33, §S.l, 31:] in 1971, simultaneously introducing the important notion of the hvector of a simplicial polytope (gbvector in the original terminology). Given the d-vector f - (f°,f1,...,fd_l), define the polynomial

f(t) -

d—l

E f t j+1 j._1 1

where again by convention we put f._1 - 1.

hm = (1 - o":

Now let

I:

Then the (d + 1)-vector h = h(f) = (ho’hl""’hd) is defined by the poly-

nomial relation d

h(t) = 2 111:1 i=0

HcMullen-Walkup write gi-l instead of 111'

When f - f(P) for some simplicial

d-polytcpe P we call h(f) the h—vectar of P and denote it by h(P) .

‘

In this

1

case we also use the natural notation h1(P) for 111' From the definition of h(f) one can explicitly write the hi as linear

conbinatiuna of the £5:

h 1 = i=0 i (d_j)(—l)i—jf d - 1 j-l’ Note that hO - 1 and h1 = f

0

— d.

0 i

min{fj(A) : £1(A) = r},

if

j < i

K(r.i.j) =

as A ranges over all simpliciel complexes. THEOREM 5 (Kruskel).

!{j+1|1+1}'

For all nonegative i,j and positive r, K(r,i,j) =

In fact, a d—vector (f 0,..., fd-l) of positive integers is the

f-vector'of some (:1 — 1)—dimensional simplicial complex if and only if

< {j+2]j+1} , fj+1_fj

< 0_jEd—2.

Hence, we have a complete characterization of f—vectors of. simpliciel complexes. Mntzkin's conjecture became a theorem with McMullen'a proof in 1970 [30; 33, Ch. 5].

Menullen did not use Kruskal's theorem; the main ingredi-

ents of his proof are (1) the existence of a particular shelling order of boundary complexes of simplicial polytapes and (2) use of the h—vector in— stesd of the f-vector.

_

.

A little bit at calculatitm shows that hi(C(n,d)) - hd_1(C(n,d)) = '

n _ d + 1 _ 1 , 1

0 S i S [d/Z].

McMullen established that for a simplicial

d-polytope P with n vettices, 0 5 111(1)) = hd_1(P) S n — (1:1

1 ,

LEE

28

0 S i S [ti/2]. The Upper Bound Theorem now follows immediately from the remark that the fj are nonnegative linear combinations of the hi' If f is in f(Pd)

UPPER BOUND THEOREM 6 (McMullen).

and

fo = n then

OEd-l.

fj Sfj(C(n,d)),

For future reference we give here the numbers fj(c(n,d)) explicitly [33,

§2.3(v1); 18, 54.7].

[d/Z]

2 n21(“;1)(j_:+1).

i=1

fj(c(n,d)) :

OSd-l. if d is even

[cl/2]

3+2 “—1 1+1 1E0 n-1(i+l)(j—i+l)’

< < _ o'j—d 1’

if d is odd

In particular.

fj(C(n,d)) «121).

0 SJ 5 («1/21 - 1,

fd_1(C(n,d)) = (n — [Ed-:11) /2]) + (u - [Ed—+d2)/2])

Even though no d—polytope has just d vertices, it is also helpful formally

to define the numbers

s (C(d,d)) % j

a (+1),

if

2,

if j - d — 1

0:55:1—2

There is an inherent difficulty in adapting McMullen's proof to triangulations of the (d - 1)-sphere; 1.2., to simplicial complexes A for which [AI is a topological (d — 1)—sphere, since not all such triangulatione are shellable [14].

However, Stanley [h2] in 1975 extended Theorem 6 to topologi—

cal and homnlogy (d — l)—spheree as well.

The method of proof seems to out-

weigh the value of the particular result, opening up many new possibilities for the interaction between combinatorics and commutative algebra. A nonempty set M of monomials Y:I---Y:s in the variables YI’YZ’ ...,‘la

if it has the property that m2 E H whenever

linzlml for some m1 6 M. A finite or infinite sequence of integers (ho’hl’h2"")

-.¢._.__m.__e_

is an order ideal of monarmiale

CHARACTERIZING NUMBERS 0F FACES OF A POLYTOPE

29

is called an 0-saquence if there exists an order ideal of monomiala M such that h1 - cardfm 6 M : deg m - i}, where deg m is the degree of the monomial Ill-

Given positive integers h and 1 define

n+1

+...+“k +1

1+1

k+1

h- 1 where

n

h- 1 +...+ “k i

k

is the i—canonical representation of h.

THEOREM 7 (Stanley).

Also put 06') = O.

Let (h0,h1,...,hd) be a (d + 1)-vector of integers.

Then the following four conditions are equivalent: (1)

(h0,...,hd) is the h-vector of some simplicial (d - l)-comp1ex A such

that for all Y E A, (ii)

§1(1kAF;Q) = 0

if i 7! dim lkAF‘

(ho,...,hd) is the h—vector of some shellahle (d - 1)—comp1ex.

(iii)

(ho, . . . ,hd) is an 0-sequence.

(iv)

ho-l,

1:120

and

(1)

05hi+15h1

foralllSiSd-l.

In (1) above, fi1(lkAF;Q) denotes the i—th reduced simplicial homology module -of the aimplicial complex lkAF computed with coefficients in the field Q of rational numbers [37].

Note the similarity between condition (iv) and Kruskal's

condition in Theorem 5.

COROLLARY 8'.

Condition (iv) holds for h—vectors of d-polytopes and of tri-

angulated topological and homology (d - 1)-spheres, and as a consequence the Upper Bound Theorem also holds for these objects.

The basic idea is that with a simplicial complex A one can associate a particular graded Q—algehra AA'

Reisner [37] demonstrated that AA is Cohen—

Macaulay if and .only if A has the property described in (1). are therefore called Cohen-Macaulay complexes.

Such complexes

Stanley explained how in this

case one can obtain an order ideal of monomials for which (ho(A),...,hd(A)) is the corresponding 0-sequence and offered (iv) as an explicit numerical . characterization of 0-sequences.

extensions see [19, 37, 41—45].

For details of this work and some of its

30 6.

LEE THE LOWER BOUND THEOREM

The determination of the minimum number of j-dimensional faces that a d—poly— tone 1’ with n vertices can have seems to be considerably harder than the cor—

responding maximum question, partly because the minima are definitely not achieved by simplicial polytopea.

Some attempts have been made to specify the

minimum value of fd_1(P)-——fo1: example, see Griinbaum [18, §10.2] and Hemllen [31]. There was, however, a long—standing conjecture as to the minimum numbers

of faces of a simplieial d-polytope with n vertices [13, §10.2].

Using the

terminology of [21] let P(d,d + 1) be a d-simplex, andrfor n > d + 1 let P(d,n)

be obtained from P(d,n - 1) by adding a pyramidal cap over one of the facets of P(d,n - 1).

Then P(d,n) is a simplicial d-polytope with n vertices.

The con-

jecture was that no simplicial d-polytope with n vertices has fewer faces of any dimension than P(d,n).

The conjecture for d - 4 was first stated by

Bruckner [12] in 1909 as a theorem, but his proof was later shown to be in— valid.

Bernette established the conjecture for j = d - 1 in 1971 [Z]

and

for 15;] Sd-Z in 1973 [5]. LOWER BOUND THEOREM 9 (Bamette).

If f is in f(P:) and f0 = n, then

.

Calling the direction of w the "down" direction, this amounts

to orienting the edges "downwards." For a vertex x of Q we define the in-dzgrea of x as the number of edges in [1 which have x as an endpoint and are oriented towards x.

Similarly, the

out-degree of x is defined as the number of edges which have x as an endpoint

and are oriented away from x.

Taking G - Q in (a) we see that

the sum of the in—degree and the out-degree of any vertex x is d.

By a k—in—stm",

(1)

k = 0,...,d, we shall mean a set consisting of a ver-

tex x of Q and k edges in Q having 1: as an endpoint and being oriented to—

wards x.

A k—out-star is defined similarly.

Denoting by Vk the number of

vettices of Q with in—degree k, we see that the number of k—in—atars in Q is

d 2

(;1 )‘r .

1-0 " j

(2)

The face-structure of Q is linked to the graphestructure by the observa— tion that there is a one-to—one correspondence between the k—faces of Q and the k-in-stars. In fact, to each k-face G there corresponds in a natural way

a k-in—stat. namely the "lowest" vertex of G together with the k edges of G

DUAL PROOF OF UPPER BOUND THEOREM which contain x, of. (a).

41

Conversely, using (b) it is not difficult to see

that each k—in—star is contained in a unique k-face G and that the k—in—stat

is 'in fact the k—in—star corresponding to G as described above.

Denoting by

E1‘ the number of k—facea of 0, it then follows from (2) that we have

‘21 ()v, d f k = H) k j

k=0.--..d

(3)

The "equations" (3) can clearly be "solved," i.e., the y 's can be ex-

pressed by the fk's.

This shows that although the definition of the numbers

Tk apparently depends on the choice of the vector w, actually

the numbers Tl: are independent of w.

(4)

Replacing w by -'w, it then follows from (1) end (4) that

ark-1114‘,

(5)

k-o,...,4

Combining (3) and (5) we obtain

fk

[d/z] j d_j = 20 [(1‘) + (1 - 6(d.21))( k )]vj.

It now remains to evaluate ‘3'

k = 0.....d

(a)

In order to do so. we introduce a k—

inc'idenae, k - 0,...,d - 1, as a pair (X,x) where X is a facet of Q and x is a vertex of X such that the number of edges in X containing x and being oriented towards x is k. vector w.)

(It is understood that we have made a choice of the

Denoting by Ik the number of k—incidences in Q, we then have

IkSv-‘rk,

k-0,...,d—1

where v denotes the number of facets of Q.

(7) In fact, each facet x of Q is it-

self a simple polytope, cf. (c), and therefore we have numbers vi,

k- =

0, ...,d — 1. associated with X in the 'same way as we have numbers Yk associated with Q.

It is clear that a vertex x of X contributes to v: if and only

if (X,x) is a k-incidence.

0n the other hand, by choosing the vector w such

that each vertex of Q not in X is "below" every vertex of x, we see that v: S 71'"

(7) holds.

Since Ik = E 7:, where we sum over all facets X of Q, we see that

42

BRENDSTED We also have

(s)

k = 0,...,d - 1

1k = (d-kM-k + (k+1)vk+1,

In fact, for each vertex x of Q there are d facets containing x, there are d edges containing x, and each facet contains d - i of the d edges, cf. (a). Therefore, a vertex x contributes to 1k if and only if either x has in—degree k, in which case the contribution is d - k, or x has in—degree k + l, ‘in which

case the contribution is k + 1.

This proves (6).

yk

-

Now combining (7) and (3) one easily shows that

(V'dik'l),

k = o,...,d

(9)

Taking

[an]

¢k(v,d) - 2

j=0

_ ._

[91) + (1 — 5(d,2j))](" a?" 1) k = 0,...,d — 2

we then see by (6) that (*) holds. To prove that (H) holds when Q is the dual of a neighborly polytope we shall prove that

Yk =(

v-a+—1 k ),

k

(10)

k = o,...,[d/2], when Q is the dual

of a neighborly polytope.

To see this, suPpose that we have strict inequality in (7)

[61/2] — l.

for some I: 5

Then there is a facet X of-Q such that v: < Yk , whence some ver-

tex x of Q not in X has in—degree k.

The out-degree therefore is d - 1:, cf.

(1).

Let G be the (d - k)—face of Q determined by this (d - k)—out—star, cf.

(1)) .

Supposing that v has been chosen in such a way that each vertex of Q

not in x is "helow’I each vertex of X, it follows that G n X - 9, and since G is the intersection of k facets x1,...,xk, we see that the k + 1 facets X,X1,

..,Xk have an empty intersection.

tope this is impossible by (d).

when Q is the dual of a neighborly poly—

This shows that for k = 0,...,[d/2] - 1 we

have equality in (7) when Q is the dual of a neighborly polytope.

(10), and so the entire proof is completed.

This implies

7

DUAL PROOF OF UPPER BOUND THEOREM

43

REFERENCES 1.

An. Bondesen and_A. Brdndsted. A dual proof of the upper bound conjecture for convex polytopes, Math. Seand. 46(1980), 95-102.

2.

P. McMullen. The maximum numbers of faces of a convex polytope, Mathe— matika 17(1970), 179-18A.

EULER'S RELATION AND WHERE IT LED G. Thomas Sallee Department of Mathematics University of California at Davis Davis, California

1.

INTRODUCTION

Let P d denote the class of all convex polytopes in Ed, d-dimensianal Euclid— esn space.

It is well-known that if P is a d-dimensionsl palytope, it satis-

fies Euler's relation

dZ

1

(-1) £10) = o

(1.1)

where £10,) denotes the number of i—dimensionel faces of P, counting 0. the empty set, to be the sole face of dimension -1 (see Griinbaum [6]; the reader may consult this source for other. standard results and definitions).

It will

also be convenient for us to write Euler's relation as:

(1.2)

I (-1)“"‘ F = 1 where the emanation is taken over all nonempty feces of a polytape P.

In the past dozen yurs, several papers [7, 14-16, 19, 20, 2h, 26, 27,

32-35] have studied functions which satisfy Euler—type relations; that is, functions which have the, property that

45

45

SALLEE

(1.3)

I H) d1“ FM?) =1 aw). again where the sum is taken over all nonempty faces of P.

Because of the

frequency with which we will use the expression on the left-and side of (1.3),

we will call it the derived function of q: and denote it as (0*(1’).

An Euler-

type relation then takes the form q>*(P) = 2MP). The most important results in discussing Euler—type relations have to do with their connection to valuations.

A valuation W mapping Pd to a vector

space v satisfies the equality

(1-4)

410’ U Q) + WP n 0) = 110’) + W(Q) whenever P U Q is convex.

(It suffices to establish (1.4) when P n Q is a

face of each [24].) Here we wish to explore the connection between the two concepts, prove some of the many corollaries of the Euler-type relation, and to call attention to the possibilities of proving many of these results by means of incidence algebras.

2.

VALUATIONS

Valuations seem to appear at every turn in a geometric context and most of the functions of interest are,

in fact, valuations.

That this occurs is not

too surprising, for the most basic functions in geometry—-characteristic functions and support functions—-are valuations.

while the latter is proved in [24].)

(The former fact is obvious,

Moreover, it is clear that if V is a

valuation, integrating W with respect to any measure will simply produce a

new valuation. Hence,

simply by checking the definitions, because the following func-

tions arise by integrating an appropriate characteristic function with respect to an appropriate measure,

(2.1) (2.2)

they are valuations:

600’), the number of lattice points inside a polytope [2]; fl(F,P), the interior angle of F in the polytope P ([6], p. 297);

(2.3)

y(F,P), the exterior angle of F in the polytope P ([6], p. 308);

(2.10) (2.5)

ym’dmk), the Grassman angles of the cone C [7]; 5(F,P), the angle deficiency of P at F [32]; and

(2.6)

Z(P), the moment vector of P

[26].

EULER'S- RELATION AND WHERE IT LED

47

Recall that the support function, H(P,u) equals sup{; p E P,u E

Sd_1}, where < , > denotes inner product. tion,

Beginning with the support func-

it is clear that the following functions are valuations:

(2.7)

m(P), the mean width of P

(2.8)

S(P), the Steiner point of P

(2.9)

mixed volumes.

[33];

[31];

This last demonstration is rather more involved

and carried out by Shepherd in [34]. (2.10)

W1(K), the quermassintegrals [8;I.

In addition, Shepherd [34] remarks that we get different valuations if H(P,u) is replaced by g(P,u) - sup{f() : x Q P,u 6 Sad} for any mono-

tone function f. Finally, there is one more consequence of the fact that H(P,u) is a valuation of a rather different flavor: (2.11)

E14, Q - (P U Q) + (P n Q) whenever P,Q and P U Q

are convex

What is the connection between all of these functions and Euler—type relations?

There are two principal results.

The first appears in [24].

while the second occurs with a somewhat different emphasis in [15], Theorems

1 and 15.

An example is also given in [24] showing that continuity is an ea-

sentisl hypothesis to the first result.

(2.12)

THEOREM.

Suppose w is a function, continuous with respect to the

Hausdorff metric, which satisfies an Euler-type relation involving faces of more than one dimension.

THEOREM.

Suppose o is a valuation on 1’(1 with the property that

«9(1’ + t) :- (p(P) + V(P,t) where V is linear in t.

where ¢k(17j) = 0 if j < k,

Then 4)

-

(2.13)

Then a: is a valuation.

) = rim

49 If P is a simple d-polytope then

for

1 = o,...,a

(3.5)

Note that the case i = d is just Euler's relation.

Perles has shown that

only [d/Z] of these equations are linearly independent ([6], p. 146). A similar set of equations hold for arbitrary functions satisfying an

Euler-type relation [15].

(3.5)

THEOREM.

Let o be a function mapping Pd to V, a vector space, such

that o*(P) = eo(P).

Then for any simple polytope, S,

1

E (-1)J[§ : 1] MM) = a 2 mi)

(3.6)

j=0

where the summations are taken over all faces of [T0012

Let F: —e any i-face of S.

S

of appropriate dimension.

Then since o satisfies an Euler-type

relation, 1

E (-1)1

1-0

i 2 mi) = we)

c 1 —F0

Sum both sides of this equation over all i-faces of 5.

Since S is simple,

each j-face is determined by exactly d - j facets and each set of d - i of them determine an i—face.

Thus, each j-fsce is counted

2 _

3-] times and

the proof is complete. E

Special cases of the result above have been known for some time [7,

20,

32].

4.

INCIDENCE ALGEBRAS

Now we wish to present an efficient framework for dealing with the types of problems which we have been considering.

It relies on the notion of an in—

cidence algebra, due to G. C. Rota [21].

While all of the details are in the

original paper, a‘brief summary is in order. Let L(P) be the lattice of faces of a d—polytope, including the empty

face, a, and P.

In the lattice, F S G if and only if F E G.

The

SALLEE

50

incidence algebra A(P) = {o : 0 maps L x L to a ring R, o(F,G) = 0 if F g 6}. Addition is defined in the obvious way and multiplication by 0 implies:

f(A) > f(0), which means w(A) >00.

So we have

' ¢(u[0,1])> 0 for all u 6 R+ and by the theory of Cauchy' 8 functional equation we obtain w(a[0, 1]) = (WHO, 1]).

Finally we get W(A)= V(A)w([0, 1]) and

further f(A) - f(0) + V(A)\y([0, 1]) and ll;([0, 1]) Z 0. AV(A)W([0,1])Z 0,

A 5 R+ , we obtain f(0)> 0.

W'(A)

Since fO‘A) = f(0) + = V(A), "'(A) = 2 and

ei-§%1,'= v([o,11)give m) - ci"W” + c6W6(A),c'1,c(')z 0. Because of W2(A) = —W' (A), w 1(A) - W'(A) for all A 6 P(E 1) we obtain

with (:1 = c'

“0’ c2

= — c'

1! 1

f(A) s c1W1(A) + c2w2(A)

and this equation holds for all A E P032) with dim A < 2 because f is in-

variant under rigid motions. wiser [2].

We can follow now the argumentation of Had.~

The valuation

5(A) = f(A) - elwlm - c2W2(A) is simply additive and by a special construction one can find (:0 6 IR with

X(A) = f(A) - coW0(A) — c1W1(A) - c2W2(A)

is simply additive and fulfills the equation

SPIEGEL

70

X((a + pm - x(aa) + X(|3A)

a.fl > o,

A e 20:2)

We obtain XOA) - lX(A) for all X 6 0+ and since every simply additive, trans-

lation—invariant, and rational homogenous valuation is additive in the sense of Minkowski, we obtain X(A X B) I X(A) + X03),

A,B 6 P012).

(See [2]. P. 63

for further details.)

V

Let S be a simplex in P012). dim A < 2,

dim B < 2,

Since 5 U S‘ - A x B with S' = -S X t and

dim S n S‘ < 2 we obtain 23((5) = 0 and so X(A) - 0 for

A E P(E2) and the theorem is proved as f(A) > 0 implies c0 2 0. El

3.

CONSEQUENCES AND OPEN QUESTIONS

First of all let us remark that the additivity in the sense of Minkowski for rational homogeneous, simply additive, translation—invariant valuations holds for all rational homogeneous, translation-invariant valuations of P(Ed) .

We

prove the following generalization of Hadwiger's result mentioned above. THEOREM 2.

Let f be a valuation of P(Ed) which is invariant under translations

and for which f().A) = AHA) hold for A 6 Paid) and X 6 0+.

f(A X B)

Then we have

- E(A) + £03) d

Proof.

With respect to [3, I», 6] we have f(AA x m!) =

Since f(tA) - tf(A)

X Arusxrsmfi). r=0 s=0

(f. E Q+), we obtain f()\A X LLB) - uX01(A,§) + “10(A,B)

and further 5(3) = X01(A.B), E(A) = X10(A,B).

X = u. = 1 presents f(A x B) -

f(A) + £03). D The question arises whether Theorem 1 holds in d dimensions.

Following

our argumentation 1n the same manner it would be interesting to know conditions where a motion-invariant valuation which is simply additive and additive in the

sense of Minkowski vanishes on P(Ed) .

The following theorem gives a possible

condition. THEOREM 3.

Let x be a simply additive valuation on P(Ed), d 2 2, which is

additive in the sense of Minkowski, invariant under rigid motions and satisfy— ing the inequality X(A) 2 a for all A contained in the unit cube H, where a is an existing real constant.

In addition, there exists a function K(Ed)_+li

NONNEGATIVE, MOTION-INVARIANT VALUATIONS

71

which is continuous with respect to the Hausdorffmetric and X + V

d

is

P0: ) monotonic.

Then x(A) = 0 for all A G P(Ed).

We mention that we make no additional assumptions for W except the con-

tinuity and the fact that V is defined on 1((Ed). Proof.

Pad).

If a 2 0 we get by usual argumentations X(A) is monotonous on

Since X04) - 0 for every cube, we obtain for A E P(Ed) and a cube W

with A C W0:

0

0 S X(A) S X(Wo) - 0 and X(A)= 0 follows.

If a < 0 and A 6 Paid) with 0 6 A E W we get -X(A)_ -a(—a Z O) and since XA 9 W for X E [0,1], we obtain setting h(\) = -X(M):

h(\ + u) = 110) + h(u.)

(Lu. 6 R+) and 1.0.) < H < =- for all x E [0 l] and so by a result of Ostrowski

[511-0) = h(1)x,x e “IR , which means xmx) - mm) for A 6 ms“) with o e AEW.

Since A 6 POE“1 )+is arbitrary, we have A - (A x (-5)) x {a}, a E A,

and 0 E A x [-a}.

As 96

(2+ such that p(A x {-s}) 5w, we obtain for X 6 11+,

X(XB(A x {—a})) = “(NA x (-a})) and it follows as X is invariant under rigid motions:

XOA) = XX(A).

This equality also holds for X = 0.

For n SIN and 51""’5n rotations around the origin gnd )‘i 2 O,

1,...,n, 121 x = 1 we define for A a mad), T(A) = KIA 1 x The transformation T : Paid) -> P(Ed) is called "Drehmittelung."

:-

x 11A “ By Hadwiger

[2] we know that for A 6 P(Ed) there exists a ball with center in the origin K and a sequence (Au)nGN in 6 = {T(A)

I T'Drehmittelung'} and $33 An - K.

By B(a;r) we denote the hall with center a and radius r. arbitrary positive real number.

Now let a be an

Because ‘0 is continuous, for P f Paid) ar—

bitrarily chosen and with r = fil- Wd_1(P), one can find real numbers r0, II with 0 < to < r < :1 and N(B(0;r1) - V(B(0;r)| < e

(i - 0,1).

Because of the theorem of Hadwiger mentioned above, the continuity of ‘4’

and the inequality r0 < r < I:1 we can find "Drehmittelungen" T0,T,Tl with the following properties:

10010) 3 [Three points do not determine a (pseudo-)plane, to be published in J. Comb. Theory, Series A], which shows that our hoped-for method of extending the theorems above to higher dimensions is doomed to failure.

Moreover, A. Mandel [private communica-

tion] has an example which shows that the generalization of the separation theorem (1') to higher dimensions is false, for a reason related to the failure of the Levi enlargement lemma. 0n the other hand, R. Cordovil [Sur un théoréme de séparation Ides matroides orientés de tang trois, preprint] has successfully used the techniques of oriented matroids to give an alternate proof of the Kirchberger

theorem (5') for pseudoline arrangements, and to generalize the Carsthéodory theorem (4') to higher dimensions.

Since his proof of Kirchberger makes use

of the Folkmsn—srence theorem on the representability of an oriented ma— troid by an arrangement of pseudohemispheres, a result which relies heavily on topological arguments of its own, it is not really a "purely combinatorial"

proof of the sort we envisioned above, i.e., a direct proof of (5); his re— sults do suggest, however, that suitably stated higher dimensional generali-

zations of (2'), (3'), and (5') may yet be obtainable.

IS THERE A KRASNOSELSKII THEOREM FOR FINITELV STARLIKE SETS? Bruce E. Peterson Dspmm: of Mathematics

Middlebw-y College Middlebuz-y, Vermont

If S is a subset of Ed and x and y are points of S, then x can see y via S

if the segment xy C S.

The set S 15 starliks if there is a point x in S which

can see every point of S via S.

The set S is finitely starlike 1f fat every

finite aubeet 1' of 5, there is a point I: of S such that t can see every point

of 1‘ via 5.

'

Krasnoselakii [2] proved that for compact sets, if every set of at most I: + 1 points can see a cannon point, then S is starlike.

Of course, then any

compact finitely atarlike net is starlike. Nonatarlike finitely starlike sets abound:

EXAWIE 1.

In E2 the region bounded by the graph of y I

1

l ' x

uymptures,

$1=i(X,y)10 SAl (K),

is {p e 321 SA1(conv(K u 12)) - A} REMARK.

We will denote this profile as y)‘.

If conv(K U P) denotes the convex hull of K U P, such convex sets

being called cap bodies (Kappenkmper) in the older literature (cf. Bonnesen— Fenchel [1, pp. 17-191), then SA1(conv(K U 2)) may be interpreted as the length of the inelastic string wrapped around K and pulled taut at P (Fig. 3). FIGURE 3

DEFINITION 3.

we call the function f(P)= SA1(conv(K U P)) the string length

function of K at P.

REMARK.

It is clear that the profile VA is simply the level curve of the

string length function. tion and, consequently,

< A} must be convex.

It can be shown that this function is a convex functhat the sublevel set I‘A - {P E R2

I

SA1(conv(K U P))

This functional approach to studying profiles is found

in Stoll [7, pp. 58—50, 54].

when the present author began studying profiles,

he was unaware of Stoll's work and developed an independent appraoch.

This

more geometric approach is outlined below, partly because it has the advantage of showing I‘A is smooth and rotund as well as convex, properties of TX

which Stall does not appear to have considered.

TURNER

90 MDNOTONICITY LEMMA.

Suppose H is any supporting line to K.

As one travels

along H in a direction away from K n H, the string length function is strictly increas ing and continuous .

Proof. K h H is either a closed line segment AB or a point A = B. If P if Q are points of H \ AB such that Q is between P and A but E is not, then we have conv(K U Q) S conv(K U P). (See Fig. A.) The strict monotonicity of the surface area functional 5A1 on compact sets in R2 (cf. Bonnesen—Fenchel

[1, p. A71) implies that f(Q) = SA1(conv(K U (2)) < SA1(conv(l( u m) - f(P). This shows that f is strictly increasing as we travel along H away from AB. The continuity of f follows from the continuity of 5A1 with respect to the

Hausdorff metric applied to the collection of cap bodies, canv(K U P). E!

FIGURE 4

EXISTENCE LEMMA.

Suppose H is any supporting line of K.

For any choice of

A > SA1(K), there exist exactly two points of 1" which lies on H. Proof.

Use the Monotonicity Lemma above and the Intermediate Value The-

orem. El MARK.

Suppose P 1 K and that conv(K U P) has nonempty interior.

set bd conv(K U P) consists of three parts.

Then the

Two of these are line segments

along the two supporting lines to K passing through 1’ and the third is the are shared by bd K and bd conv(K U 1’).

(See Fig. 5.)

Since bd K n bd canv (R U P)

is compact, this shared are is closed with two distinct endpoints.

Since

conv(K U P) has nonempty interior, its boundary is a simple closed curve which may be oriented in the usual counterclockwise sense.

Let A denote the end-

91

CONVEX CAUSTICS FOR BILLIARDS

point of the shared are which is prior from P in this sense and let 2 denote

the other (later) endpoint.

Then A and Z are the first and last points, re—

spectively, of bd K‘vhich touch the string of length A wrapped around K and pulled taut at P. DEFINITION 10.

This intuitive picture motivates the following definition.

Let P

+ In: — zll

where A and Z are the lead and trail contact points 'of P, respectively.

CONVEX CAUSTICS FOR EILLIARDS Proof.

95

We refer to Stoll [7, p. 60], or to Turner [10, pp. 185-194] for

a more detailed proof.

REMARK.

Since F%_E_%fi and i%_f_%i are unit vectors, their sum grad f(P) lies

on the bisector B of the string angle APZ.

See Fig. 9.

By Theorem 1, FA is

smooth, sodwhen its boundary YA is parametrized by arclength s, as yA(P) exv ists, andd —— 7H(P) is a unit tangent vector lying along T.

Since B and T

are normaldlines, i— yA(P) and grad f(P) are orthogonal, as they should be. For the unit tangent vector of a level curve of a C1 function f must be orthogonal to grad f.

THEOREM 2.

Suppose K is any compact convex set in R2 contained in the in-

terior of a smooth compact convex body T E R2.

Then K is a convex caustic

for billiards inside r if and only if bd F is an Rz-embedded profile of K.

Iraoji A > SA1(K).

(é) Suppose bd r is the Rz—embedded profile of K determined by Then I‘ = I‘A.

Suppose a billiards trajectory travels along a

“Gm 9

ma n?) ’I P-A

/ /

, I P up-Au I

/ A / I

supporting line to K before reflection at P E bd TA = VA. The unique supporting line T to FA at P is the normal to the bisector B of string angle APZ by Theorem 1. Let C be a point of B n T+ \ P. The angle of incidence is thus a complementary angle of angle AFC, as in Fig. 10. After the reflection at P, the angle of reflection must also be a complementary angle of angle AFC. But

96

TURNER

FIGURE 10

since B bisects angle APZ, the measure of angle AFC is equal to the measure of angle ZPC. ZPC.

Hence the angle of reflection must be a complement of angle

This forces the billiards trajectory to travel along ray PZ after re-

flection at P.

Since ray PZ E supporting line P2 of K, we have shown that K

is a convex caustic for billiards in I“. (‘0

Suppose K is a convex caustic for billiards in I‘.

Since I‘ is a

compact convex body in R2, bd I‘ may be represented as the image of s recti-

fiable simple closed curve which may be parametrized by artlength this boundary curve as v.

s.

Denote

Then 1: [0, SA1(I‘)] + bd I‘ is a (11 function of its

parameter 5, since I‘ is smooth and J is a unit tangent vector which lies along the unique supporting line 1' toSF at 1(3).

Consider the composed func-

tion

f(v(S)) = SA1(conV(K U 1(8)» Since f is also a Cl function by Stoll's proposition, 3—5 f(y(s)) exists and equals grad f(y(s)) -

5Y(s) by the Chain Rule.

We claim that for all s E

[0, SA1(1")1. grad f(Y(S)) K is a convex caustic for billiards in I‘ by hypothesis.

Given 1(a),

draw the supporting lines to K and let A(s) and 2(3) denote the lead and

trail contact points of 1(5), respectively.

The lav' "angle of incidence e-

CONVEX' CAUSTICS FOR BILLIARDS

97

quals angle nf reflection" at y(a) implies that

g_

d5

s

(See Fig. 11.)

d_y_ d8

-As

"11(8) -A(S)II

=_fl_

ds

s

—Zs

|Iv(s) '10:)"

Hence

1(52-Ags) +s()=1§)—Zs] “1(8) - A(S)II+I|*1(8) - 2(5) ||

0 .

By Stall ' a proposition ,

1(5) — Ms) M grad f(v(5))' Ms) _ M5)" + ||y(s) — 2(5)“ Hence ST: £(y(s)) - 0 as claimed and f(y(s)) E A (constant). all s E [0, SA1(I‘)] and hr] 1‘ E Yx‘

bd I‘ cannot be a simple closed curve.

FIGURE 11

50 y(s) E y)‘ for

It is clear that y)‘ E ha 1‘, for otherwise Thus Y). = bd I‘. D

Yul-2(a)

llY(a)-Z(s)ll d: d1

31 s E -A( 5 1 [Ms -A(u)ll

’r

HISTORICAL REMARK.

Theorem 2 is a generalization of a statement (made without

proof) concerning ‘convex caustics for billiards with differentiable boundary

98

TURNER

curves found in Sinai [6, p. 87].

The author found this statement interesting,

for it asserted that a certain metric invariant (which we have interpreted as 5A1 or conv(K U P) here) was related to a reflective property; namely, posses— sion of a convex caustic.

The author is indebted to his advisor, Professor

Walter H. Gottschalk, for pointing out that the metric invariant could be in— terpreted as the length of a closed loop of inelastic string wrapped around the caustic.

This interpretation led to the conclusion that the smoothness

assumption on the boundary of the caustic was probably unnecessary, and it suggested, of course, the close connection between profiles and ellipses which the reader has seen us exploit in several ways.

AN OPEN PROBLEM.

In one sense, Theorem 2 provides a complete answer to the

search for convex caustics in the plane:

given any compact convex set K, K

is a convex caustic for billiards inside any member of its one parameter fam—

ily of profiles (VA I A > SAl(K)). not very satisfactory.

In another sense, however, Theor- 2 is

If we are given any smooth compact convex body 1‘,

how can we tell whether 1‘ possesses a convex caustic K for billiards inside I"!

By Theorem Z,we know that bd I‘ must be an RZ-embedded profile of K, if

K exists, but we do not know how to distinguish profiles intrinsically from boundaries of general smooth compact convex bodies in R2. a necessary condition; namely, 1‘ must be rotund.

Theoran 1 provides

But it seems highly unlikely

to this author that this condition should also be sufficient.

Unfortunately,

a "counter-ample" to demonstrate this nonsufficiency fell through as this

paper was being readied, so it is still possible that rotundity 'is sufficient. The problem, then, is to find some converse to Theorem 1.

This problem has been partially addressed by the Russian mathematician

V. F. Lazutkin in [3], where he states the following theorem.

THEOREM (Lazutkin).

Suppose that I‘ is a smooth compact convex body in R2.

Let p(s) be the radius of curvature function of a boundary curve of 1‘ para—

metrized by arclength e.

If 0 < m < p(s) < M (m for all s E [0, SA1(I‘)] and if [1(a) is C553, then there exists a discontinuous family of convex caus— tics for billiards inside X' with boundaries contained in a small neighborhood

of bd r. Proof.

'

.

We refer the interested reader to Lazutkin's paper [3].

CONVEX CAUSTICS FOR BILLIARDS REMARK.

99

It is easy to see that the profiles of a simplex have a discontinuityr

in their curvature function wherever they cross an extended side of the sim-

plex.

Thus there are elementary examples of profiles where 0(a) is not even

continuous! essary.

This shows that Lazutkin's C553 condition is very far from nec—

We note, however,that 0 < p(s) < M < ” insures that F is rotund,

since otherwise 9(a) is unbounded.

3.

CONVEX CAUSTICS FOR BILLIARDS IN R3 We would like to describe some of our recent progress in the search for

convex caustics for billiards in R3. By a supporting line of a compact convex set K in R3, we will mean a line L contained in a supporting plane of K such that L n K f b.

Then Defi-

nition 1 above needs only a trivial modification to describe a convex caustic

for billiards in R3. Our first guess in searching for caustics in R3 was that the proper reflective container for a caustic would be defined by capbodies with constant two-dimensional surface area.

DEFINITION 5.

Thus we investigated the following objects.

Suppose that K is a compact convex set in R3 with two dimen-

sional surface area, SA2(K).

Then the Ra-ambedded profile of K determined

by the constant A, where A > SA1(K), is {P E R3 I SA2(conv(K U P)) - A}.

We

will use YA again to denote such profiles.

EXAMPLES.

Unlike the planar case, there are no classical examples of R3-

embedded profiles, although they were studied by Stoll in [7, PP- 48-50, 55]. Let us briefly construct some elementary examples. Let K be a singleton point.

Then for all P E R3, conv(K U P) is a closed

line segment and SA2(c;nv(K U P)) - 0 I SA2(K).

vi = ‘-

Thus for any A > SA2(K),

Recall that R —embedded profile of K was a circle centered at K with

positive radius.

Thus profiles of K depend strongly on the choice of the

embedding space. Now let K be a line segment Fl.

If P E line F1F2, than conv(K U P)

is a closed line segment contained in this line, so SA2(conv(K U P) = 0 again.

Hence the line F F does not intersect any RJ—embedded profile of K. If 1 2 P E R3 \ line FlFZ' thenconv(K U P) is the relative interior of the triangle

TURNER

100 PFl.

Let MP} denote the orthogonal projection of P — F1 on F2 — F1.

SA2(conv(K u 2)) = 2% “Fl — F2“ - I|N(P)II .

Then

(See Fig. x2.) Thus for x >

“200 = o, "F1 _ F2“ , a constant} - + — {1’ E R 3 I IIN(P) ll _ Y). _

Hence Y). is a circular cylinder with line F1172 as the central axis and radius

[IF equal to 4

— 172”

Recall that

We note that in this case YA is unbounded.

l the Rz-embedded profiles of segment 1711’

2

were ellipses.

Let K be the disk lying in the xy-plane centered at the origin with redius a > 0.

Then conv(1( U P) is, in general, a solid oblique circular cane.

Let P = (x,y,z).

Then it can be shown by using some calculus that

21!

SA2(c'onv(K U P)) = §J «(a + M + y2 cos a)2 + :2 as + 1ra2 0 Hence given A > 2132 = SA2(K), 2w VA = {(x,y,z)

a2}

"(a + I/}(2 + y2 cos a)2 + 22 d6 = A —

I g!

0

Thus V)"

in this case,

gral equation.

is represented as the solution of an elliptic inte-

'

Finally, let K be a closed ball in R3. If P e R3 \ K, then conv(P U K) is a "dunce head," i.e., the convex hull of a right circular cane with vertex

FIGURE 12

CONVEX CAUSTICS FOR BILLIARDS

101

at P and sides tangent to K perched upon K.

It should be clear that if \ >

SA2(K). then YA will be a concentric sphere properly containing K.

At last,

here is an example of an R3 -embedded profile of K which clearly exhibits the

reflective property of the R2-embedded profiles; namely, the inner ball K is

a caustic for billiards inside its outer concentric ball, conv VA'

From a billiards point of view these examples of R3-embedded profiles are disappointing.

Except for the last example, it can be shown that K is

not a caustic for billiards inside conv VA'

Consider,

segment Fl once again with VA a circular cylinder.

for example, the line

The tangent plane to

P 6 YA is a plane normal to the radius vector of the cylinder at P.

Thus a

billiards trajectory which passes through F1 and strikes YA at a point P radially above F2 is reflected away from segment F F towards the point F' E .\ 1 2 1 line 1?l which is symmetric to F1 with respect to F2. Thus this trajectory travels along a supporting line of segment Fl before reflection, but is not reflected along a supporting line, since PF 1' n K = w. A little reflection (no pun intended) should convince the reader that the proper reflecting container for the line segment FlF21s a (prolate) e1— lipsoid of revolution using line Fl as the axis of revolution for any :1lipse focused at F1, F

This example reveals that the search for caustics

in R3 leads to consideiation of nonprofiles in R3.

This conclusion is fur-

ther buttressed by the following observation (suggested by a colleague, Burt if K is the disk lying in the xy-plane centered at the ori-

Cassler at LSU):

gin with radius a > 0, then K is a convex caustic for billiards in an (oblate)

ellipsoid of revolution.

Choose (-s,0,0) and (a,0,0) as foci of an ellipse

in the xz-plane and revolve this ellipse about the z-axis.

A rather long

computation using elliptic coordinates convinces one that the disk is indeed

a convex caustic for billiards inside the convex hull of such an ellipsoid. The reader may have already noticed that every choice of K that we have used

in our examples above is a (degenerate) ellipsoid.

Some further thought in

this direction led to the formulation of Theorem 3 below.

DEFINITION 6.

A quadric surface S is said to be confbcal to a given ellipX2

22

—— = 1) if there exists same A E R such that a2 +§§-+:2 soid E ' {(X.y.z) I—

-

x2

-

2

s={(x.y.z)lm+b—2%+ m‘ 1}

102

TURNER This definition is standard and may be found, for example, in Salmon

REMARK.

[5, p. 166]. If we choose the positive sign in front of A2, then clearly 5 must be another ellipsoid.

If we choose the negative sign in front of A2,

however, there are three possibilities. If A2 < min{a2,b2,c2}, then s is still an ellipsoid.

Supposing that c2 = min{az,hz,c2}, if c2 0, the k-fiflee alignment on a set X con-

sists of all subsets of k or fewer points ofX together with X itself. k-free alignment onnpoints will be denoted Fk(n). Fk(n) = F(n).

The

Evidently, for k > n - 1, .

129

I

A PERSPECTIVE 0N ABSTRACT CONVEXITY

For integers n > k > 0, the quasicircuit Qk(n) is defined to be the slignment on an 11 point set in which each subset of n — 2 or fewer points is convex and exactly k subsets of n — 1 points are convex [32]. %(n) = F(n).

Qn_1(n) may be visualized as

simplex together with an interior point. in the matroid sense:

Evidently,

the vertices of an (n — 2)—

Qo(n) I Fn_2(n) is a true circuit

it is the unique matroid of rank n — 1 on n points.

For each k, the k-free alignments form a variety (k-free). of all free alignments forms a variety (free).

The class

The quasicircuits together

with the free alignments form a variety, but the main interest in the quasi—

circuits is the role they play in characterizations by forbidden subspaces (cf. Section 7).

CONSTRUCTIONS HITH ALIGNMENTS AND VARIETIES

5.

The list of examples of alignments can be further augmented by means of several natural constructions.

In terms of varieties, we shall be interested

in knowing what varieties are closed under which of the constructions. summarizes some information along these lines.

of subspaces was introduced, and,

Table I

In Section 3 the construction

by definition, every variety is closed

under this construction .

TABLE 1.

Variety

Closure Properties of Varieties

Product-

Sum—

closed

closed

Join— closed

Contractible

Lattice— closed

Principal

(Section 6C) (P0)

no

yes

no

no

no

(TO)

no

no

no

no

no

yes

(monotone )

no

no

no

yes

no

yes

(matroids)

no

yes

no

yes

no

yes

(ED)

no

yes

yes

yes

no

yes

So. 51,

52,

(Carath < n) (rank < n)

(ls-free)

S3

yes

yes

yes

yes

no

no

yes

no

yes

no

yes

no

yes

no

no

yes

yes

no

yes

no

only for k - o

only, for all I: k _ w

yes

all k

JAMISON-WALDNER

130

A.

Products, Sums, and Joins If (x,1:1) and (31,1:2) are aligned spaces, then

£1 81:2 = {L x M:

1.6.51 andMEIZ}

is an alignment on x x Y (cf. [15, 29, 36, 48, 51, 52]). 131 8L2) is the praahct of

(XJII) and

(Y'£2)'

The space (X X Y,

Products of any number of

factors can be formed similarly by first forming the collection of all "boxes" ——cartesian products of one convex set from each factorT-and then taking the

alignment they generate.

(In general, the boxes themselves form an align-

ment only if the number of factors is finite [29].) Note that a product of d copies of the real line R does not yield the

ordinary alignment on Rd, but rather the box alignment on Rd in which the only "convex" sets are boxes with sides parallel to the coordinate axes. This alignment has been studied extensively by Eckhoff [15] and others [36, 48, 52]. other structures fare better under products.

The geodesic alignment on

a cartesian product of graphs is the product of the geodesic alignments on

the factors [37].

The order lattice alignment on a product of lattices is

the product of the order lattice alignments on the factors.

Since the or—

dinary alignment on R is an order lattice alignment (for the usual order),

the box alignment on R“1 should also be an order lattice alignment.

Indeed

it is, with Rd made into a lattice by coordinatewise ordering. Using the finitary property, one can show that a variety is closed un— der the formation of arbitrary products provided it contains the product of

each pair of its finite spaces.

Such a variety is said to be product-closed.

The varieties (so), (51),. (52), and (53) are product—closed.

As we shall

see (Section AC), being product—closed is a strong requirement. A less demanding notion is that of sum.

The sum of a collection of a—

ligned spaces is obtained by representing them on disjoint underlying spaces

and then aligning the disjoint union with all possible unions of one convex set from each summand.

A variety is awn—closed if it is closed under the

formation of arbitrary sums.

Again it is sufficient, by the finitary prop-

erty, that the sum of each pair of finite spaces in the variety again be in

the variety.

Being sum-closed is a weak condition, enjoyed by most of the

varieties we have discussed.

.

‘

A PERSi’ECTIVE 0N ABSTRACT CONVEXITV

131

If (£1), 7 E F, is a collection of alignments on a set X, then the small— est alignment R on x containing all £7's is the join of the .Cy's in the lat— tice of all alignments on X. It can be shown [29], that if R is the join of

(£7), 1 e 1', then

(5.1) «(17.) = n £703) .7El‘ for all finite subsets E of X. If this formula actually holds for all subsets E of X, we shall say that R is a strong join of (£1), 7 E r. Further, (5.2)

The ordinary alignment on any real vector space is a strong join of

total order alignments.

(5.3)

The order alignment on any partially ordered set is a strong join of

total order alignments.

Fact (5.2) is essentially a.version of the separation properties of ordinary convex sets. (5.3) is a consequence of the fact that any partial order, as a set of ordered pairs,

is the intersection of the total orders extending it.

The occurrence of strong joins is not a coincidence, but an illustration of

a general principle involving varieties.

(5.11)

If an alignment is a join of alignments from a variety V, then it is

actually a strong join of alignments from V.

while it can be shown that the join of any finite number of alignments is strong [29, p. 20], this need not be the case for infinite joins.

Indeed,

the free alignment on an infinite set X is the join, but not the strong join,

of the k—free alignments (k = 1, 2, 3, ...) on x. A variety is join-closed if it is closed under the formation of arbitrary joins.

Again, for a variety to be join—closed it suffices for it to

contain pairwise joins on finite spaces.

B.

The category of Aligned Spaces and Minors The class of all aligned spaces can be made into a category ALN in which

the-morphisms from (X,£1) to (Y,£2) are maps from X to Y such that the pre-

132

JAMISON-WALDNER

image of each convex set in Y is convex in X.

The notions of product and

sum introduced above are the correct ones for this category.

The category

ALN also possesses inductive and projective limits. (5.5)

Any variety is closed under arbitrary inductive and projective limits.

Since by the finitary property, every aligned space is the inductive limit

of its finite subspaces, closure under inductive limits is essentially an equivalent formulation of variety axiom V3.

The category ALN also contains quotients.

For the most part, these are

not well—enough behaved to play a major role in the structure theory of alignments.

There is, however, one type of quotient that is of general interest.

If K is s convex set in an aligned space (X,£), then the contraction of (X,£) over K has X \

K as underlying set and

II K= (L\ K: as alignment.

LEI andKEL)

A subspace of a contraction of X (or, equivalently, a contrac-

tion of a subspace) is a minor- of X. theory via matroids [1b, 58. 63].)

(This term is borrowed from matrix

It is not difficult to check that a minor

of a minor of X is again s minor of X.

A variety closed under the formation

of arbitrary contractions (and hence of minors) is cmtmctible. As with products, sums, and joins, there is also a finite sufficient

condition for contractibility .

(5.6)

A variety V is contractible lit for each finite space (E,£) in V and

each 1: in E, the contraction of E over £(p) is in V.

c.

Product-Closed Varieties The relative strength of product-closure is indicated by the results

below.

(5.7)

Any product—closed variety is join—closed.

(5.8)

A product-closed variety other than (1 pt) or (0—free) is sum—closed.

133

A PERSPECTIVE 0N ABSTRACT convsxm (5.9)

The only product-closed variety containing both Q°(2) and 01(2) is

(all), the Variety of all aligned spaces.

(5.10)

The only varieties closed under both products and contractions are

(1 pt), (D-free), and (all).

As an illustration of some of the techniques involved, we shall actually prove these results .

Proof of (5.7).

The join of two alignments £1 and £2 on a space X is

isomorphic to the diagonal subspace of x x X with the product alignment 1:; 8

£2. D As an aid in

‘ (5.11)

(5.8) and

(5.10), the following lemma is useful.

If V is a product-closed variety other than (1 pt) or (0—free), then

v contains (22(2)’ - 17(2). Proof.

Since any 0—free alignment is a subspace of products of 00(2),

V is not contained in (0-free) or it would be either (1 pt) or (O—Eree). Hence V contains a space X with a nonempty, proper convex set R. x E K and any y E X \ X.

Pick any

Then {x} is convex in the subspace {any} of X, so

this subspace is either 01(2) or 02(2). In the latter case we are done.

In the former, V contains 01(2) which

can be represented in two ways on {0,l}--one with 0 convex, the other with l convex.

The join of these representations is Q2 (2), which by (5.7) must

be in V as desired. El

Proof of (5.8).

Given spaces x and Y, pick and fix p E x and q E Y.

The sum of X and Y is then isomorphic to the following subspace of the prod—

uct X X 02(2) X Y:

(X x {0} X (11)) U ((1)) x (1} x Y)_ D Theorem (5.9) is an immediate consequence of the following stronger result:

JAMISON-HALDNER

134 (5.9')

Any aligned space (X,£) is isomorphic to a subspace of a product of

copies of 00(2) and (21(2)h'aof.

Take 1‘ = x U .6 as index set and let Z = 7%. (0,117 where the

factors indexed by points in X are taken isomorphic to 00(2) and the factors indexed by sets in 13 are taken isomorphic to (31(2) with {1} convex.

For each

x in X, define 45" in 2 as follows:

faryexgr,

forKE-CEI',

The factors 00(2)

one—to—one.

¢x(y)-1

1fx=y

¢X(y)=0

ifxry

$(K)=1

ifxEK

x ¢X(K)=0

ifxe‘K

for each y in X ensure that the map x -> ¢x from X to Z is

The factors (21(2) for each K in I ensure that it is an isomor-

phism with s subspace of Z. El Proof of (5.10).

Suppose V is a contractible, product—closed variety

other than (1 pt) or (0-free). alignment on (0.1}.

By lemma (5.11), V contains 17(2) as the free

Now V contains the product F(2) x P(2).

product over the convex point (0,0). contains (0.1), but not vice versa.

Contract this

In the contraction, the hull of (1,1) Hence (11(2) is a minor of'F(2) X F(2)

and thus is in V. Now contract the convex edge ((0,0), (0,1)} of F(2) X 17(2). is (20(2) which must then be in V.

This space

Since V contains (10(2) and 01(2), V -

(all) by (5.9). D On the basis of these results one might suspect that there are really rather few product-closed varieties.

On the contrary, it can be shown that

the lattice of product-closed varieties contains a copy of a complete Boolean algebra on a (countably)

infinite set.

It would be of some interest to ob-

tain a complete description of the lattice of product—closed varieties.

All known product-closed varieties are defined by some type of "separation" axiom.

Is this possibly always the case for some reasonable notion of

"separation axiom? "

135

A PERSPECTIVE 0N ABSTRACT CONVEXITV

D.

Join-Closure of Varieties .

For any two varieties V and W, let

V - N be the class of all aligned

spaces (X,£) where 1.’ is the join of an alignment in V with an alignment in

H. We set v2 - V - V and define analogously Vu to he the class of all alignments obtainable as the join of n (or fewer) alignments from V.

Likewise,

Va denotes those alignments which are joins of some (maybe infinite)

tion of alignments in V. yield varieties.

collec—

It can be shown that all of these constructions

(Axioms V1 and V2 are trivially satisfied; the difficulty

lies in establishing V3.) Thus Va is the join—closure ofV (i.e., it is the smallest join-closed variety containing V).

of particular interest is the join-closure (TOW.

As already noted in (5.2) and (5.3) the ordinary alignments on real vector spaces as well as all (partial) order alignments belong to no)".

This sug-

. gesta that a study of (Tofu would be a reasonable step toward a characteri— zation of the finite subspaces of Euclidean space.

This approach has in fact

been taken implicitly by Goodman and Pollack in their studies of the geome— try of "allowable sequences."

(See [18] and Pollack's chapter in these Pro—

endings.) While ('I‘O)“I consists of those spaces determined by "two-sided" total orders, Mon)“1 is the related class determined by "one-sided" total orders. Although no simple characterization of Go)” is known, the join-closure of the monotone alignments is the variety of antimatroids [33, 38]:

Mon)" = (ED) Another simpler result along these lines is

(rank < 1)” - an) Indeed, any maximal chain in any alignment is a rank 1 subalignment, and every alignment is certainly the join of its chains. Notice that with every variety V we have a chain of varieties Vn S Vn+1 whose supremml (in the lattice of varieties) is the join—closure V”.

These

chains of varieties 'provide abundant additional examples of dimension parame— ters:

jv(X) = minin:

x E V“)

JAMISON-HALDNER

136 6.

RELATED VIEWS:

LOGIC, LATTICES, AND ALGEBRAS

A.

Syntax of Varieties In Section 2 alignments were introduced in two ways:

and as hull operators.

Also,

as systems of sets

two ways of viewing varieties were noted:

classes of spaces and as downsets in the ordered set FinALN.

as

Loosely speak—

ing, these views are semantic in that they directly involve specific structures rather than the logical forms of the properties defining them.

Here

we will briefly take a syntactical view, giving attention to the logical structure of properties of alignments.

The exposition is simply a working

out of well-known principles in the first order logic of relational structures for the case of alignments.

Details may be found in the standard ref-

erences [5, 6, 19]. To know an alignment J! on a set X, it is necessary and sufficient to know for each subset S of X and each point p in X whether or not the rela-

tion p E £(S) holds.

In view of the finitiary property (2.1), it suffices

to know this when S is finite.

Hence, we may associate with any aligned

spacer (x,£) a system 9 = (mn(.ll))n=l of n + 1—aty relations (i.e., subsets of

xnfl) defined by (2, x1, ..., X“) E mn(£)

iff

251(21, ..., x“)

These relations uniquely determine the alignment and enjoy the following properties:

ARSl.

(2,1) 5 ml for all z

ARSZ.

if (2, x1, ..., x“) E m“. then for any permutation 1r of l, ..., n,

ARSB.

if (1, x1, ..., x“) E Lun and xn_1 = x“. then (2, x1, ..., xn_1)

(z, x1r(1)' ..., x«(n)) E ”n

E m ARSA.

n-l

if (2, x1, .... X“) E In“. then for any y, (z, x1, ..., xn, y) E

I”n+1 ARSS.

for any n x 11 matrix of elements (xij) and elements yl,...,y':I

and 2, if (yi, x11, x12, ..., xi“) E mn for all j and (z, yl,

---, yn) 5 u“, then (z, x11, ..., x11, ..., xnn) E mnz

A PERSPECTIVE 0N ABSTRACT CONVEXITY

137

Conversely, any sequence of relations n = (tun) satisfying ARSl—ARSS arises as Mn“) for some (unique) alignment £-—juat interpret (2, x1, ..., x“) 6 mn u 2 is a convex combination of x , . . . , x n and define the alignment to mean

of convex sets accordingly.

Thus alignients maynbe identified with rela—

tional structures satisfying ARSl-ARSS. For convenience, we shall henceforth drop the tun—notation and write sim— ply z E £(xl,

..., xn).

The logical formulas with which we shall be concerned

are those obtainable from atomic formulas of the kind 2 E £(zl, ..., x“) and x - y by using logical connectives A (and), v (or), F (not), ” (implies). For example, separation axiom So can be stated as

so:

VxVy

xE£(Y) AyEJ-‘(X)’x=y

Likewise, spaces with Helly number < n are those satisfying

Hn:

, Yxo, x1, ...,xn3p v

where A denotes "omit."

n . /\ pE£(x°, ...,fii, ...,xn) i=0

(Hn actually says any n + 1 points have nonempty

core [30, 36, 53].) Both of these statements are first order—-that is, quantified over ele—

ments only and not over sets. does not.

Theorem V1.2.8 of

(6.1)

The first defines a variety and the second

This is a consequence of a general result of Robinson and Les (see [6]) which for alignments states

Varieties of alignments are precisely the universal classes A, that is,

classes defined by universally quantified first order axioms. This may not always appear to be the case. (Section 43)

For example, the exchange law

appears to be quantified over sets and separation axiom 53 (Sec—

tion AC) appears to postulate the existence of hemispaces.

But universally

quantified first order equivalents can be found without much trouble. reader may wish to try his hand at this.

The

As a rule of thumb, assertions

about all convex sets may be reduced via the finitary property to assertions

about polytopes.

The "general" polytope can then be treated as the hull of

a finite number of variables. - than just one.

The result may be a sequence of axioms rather

As an illustration, here are axioms for the variety (Csrath

< n)--we take all axioms (Zk with k > n:

JAMI SON-HALDNER

138 ck:

Vxl, ..., kp

pE£(x1. ---, xk)

k a v pE£(x1, i=1

ii,

xk)

Using the fact that varieties can be defined in terms of minimal forbid—

den subspaces (Section 3), it is easy to see why (6.1) holds. pose (E,£) is a finite aligned space.

Indeed, sup-

The requirement that a space contain

E as a subspace can be axiomatized by taking one variable x1 for each point of E and postulating

(E EX):

'

3x1, ..., xn

(_/\ xi 34 x1) A Mxl, ..., x“) 1741

where 4* is a long conjunction of all relations of either the form 2 E £(F)

or the form 2 Q £(17)——according to which holds in E.

The negation of (E E X)

is universally quantified and states that E is not a subspace of X.

Thus

any variety can be defined by a list of axdoms, one for each minimal forbidden subspace.

Since any finite number of axioms can always be combined,

by conjunction into a single axiom, we obtain

(6.2)

A variety has only finitely many minimal forbidded subspaces iff it

can be defined by a single universal first order axiom.

A further illustration of the important correspondence displayed in

(6.1) and (6.2) between semantics and syntactics is the well—known fact (see

Theorem v1.1.3 0': [6]) that product-closed varieties can be defined by Horn sentences.

It would be interesting to have similar syntactical descriptions

of join-closed varieties and of contractible varieties.

B.

Universal Algebras We have just seen how alignments can be viewed as a type of relational

system.

They are also closely related to another, widely studied type of

relational structure: can be found in [16].)

universal algebras

[6, 19].

(A concise introduction

Universal algebras'are those relational systems in

which all the relations are functions, that is, operations on the underlying

set.

Rings, groups, lattices, semilattices, and vector spaces over a fixed

field are examples.

7

A PERSPECTIVE 0N ABSTRACT CONVEXITY

139

For any universal algebra, the family of subalgebras (subsets closed under all the operations)

forms an alignment, and it is wellknown that any

alignment can be represented as the subalgebras of some algebra structure on

the underlying set (hence the traditional description of alignments as "al— gebraic closure systems").

However, there is no canonical way to assign an

algebra structure to a general alignment, so it is difficult to carry over

information from the theory of universal algebras into the general theory

of alignments.

Nonetheless, many important specific classes of alignments

(affine, semilattice, order lattice) arise from important classes of universal algebras, and this author believes that more use could and should be made

in "axiomatic convexity" of examples from universal algebra. There is also a well-developed and highly successful theory of varieties of algebras.

(See [16] on this point.)

is an equational class, defined by axioms

Syntactically, a variety of algebras that are simple equations of poly-

nomials (compositions) in the basic operations.

Semantically, varieties of

algebras are classes closed under the formation of subalgebras, products,

and quotients. Although it would be nice to emulate the theory of varieties of algebras, one cannot use for alignments the same semantic requirements.

We have al—

ready seen in (5.10) that there are only three classes of alignments closed under subspaces, products, and contractions, hardly a discriminating classi—

fication scheme.

A major difference lies in the fact that subspaces are far

more general than subalgebrus--any subset can be a subspace, but subalgebras correspond to convex subsets.

For a discussion of ordinary convexity from the universal algebra view— point, see [A7].

c.

Algebraic Lattices and Lattice-Closed Varieties Every alignment as a family of sets is a lattice under set inclusion in

which infimum is intersection and supremum is convex hull of the union.

has long been known exactly which lattices arise as alginments. 8 and B' on page 187 of [2].)

It

(See Theorems

They are those lattices which are complete

(arbitrary sups and infs exist) and algebraic, defined as follows.

An ele-

ment c of a lattice L is compact iff whenever c < sup 5 for some S E L, there

is a finite F E S with c < sup F.

a sup of compact elements. ‘ are precisely the polytopes.

A lattice is algebraic if each element is

Note that the compact elements of an alignment

140

JAM! SON-HALDNER TABLE 2.

Lattice Properties of Varieties

complete algebraic

all alignments SI:

atomic

points convex

matroids

upper semimodular

antimatroids (ED)

dually locally distributive

(see Chapter 7 of

[8]) (Carath < 1)

distributive

(rank < n)

breadth < n

It has long been popular to classify and identify various types of alignments by properties of their lattices.

(See especially [40].)

Table 2 con-

tains several classes of alignments and their characteristic lattice properties.

(Note that (Carath < 1) is precisely the class of topologies which

are also alignments.)

when points are convex,

this classification of align-

ments by their lattices is quite satisfactory since S1 spaces are determined up to isomorphism by their lattices of convex sets. when points are not convex, the correspondences become ambiguous.

Non-

isomorphic—and in fact geometrically very different—-aligned spaces can have

isomorphic lattices of convex sets.

As shown by (6.6) below, every finite

aligned space can be realized as a subspace of an alignment lattice-isomor-

phic to a free alignment. done

(e.g.,

One could eliminate the ambiguity-—as is often

[65])-—by requiring points to be convex.

But this also elimi-

nates some pleasing structural considerations, such as the characterization of antimatroids as joins of monotone alignments (Section 5D) and the simple forbidden minor characterization of matroids and antimatroids

(Section 7).

It seems best to recognize the validity and usefulness of both classi—

fication methods for alignments:

by lattices and by varieties.

in fact, an interesting area of overlap.

There is,

A variety V is lattice—closed if

for each space (X,£1) in V, every space (Y,£2) with £2 lattice-isomorphic to £1 is also in V.

Since the variety (ht(x) < n) was defined in terms of

the lattice of convex sets, it is certainly lattice—closed.

By (4.1.111),

the variety (rank < n) can be defined by a lattice property, so it is also lattice—closed.

servat ion .

Further examples arise from the following simple simple ob_

A PERSPECTIVE ON ABSTRACT CONVEXITY (6.3)

141

If V and H are lattice-closed, then so is V - N. The next results indicate some restrictions on lattice-closed varieties.

(6A)

There are only two varieties that are both lattice—closed and join-

cloeed, namely, (O—Eree) and (all).

(6.5)

If V if (all) is a lattice-closed variety, then there is an N > 0 such

that rank(X) < N for all X in V.

'

This second result follows at once from (6.6) below and the observation

that a space of rank > n contains the free alignment F(n) as a subspace. (The relative alignment on any independent set is free.)

(6.6)

Suppose each point of the aligned space (X431) lies in all but finitely

many £1—convex sets.

Then (X,J.‘1) can be embedded as a subspace of a space

(LIZ) where .62 is lattice-isomorphic to a free alignment.

That is, as a

lattice, £2 is a complete Boolean algebra.

Proof of (6.6).

Take Y - X U £1.

o(x)={LE£1:

x¢ I.)

For each point x in X, let

Align Y with families M E Y such that

xEM

iff

for each x in X.

60094 Since (’0!) is finite for each x, the collection .52 of all

such M is an alignment on Y.

Since for each subfamily R E £1,

szm) - a u (x: m.) gm one can show that £2 is isomorphic as a lattice to P011031).

It is also not

hard to see that 131 - £2 | X, as desired. CI The hypothesis of (6.6) admits all finite aligned spaces, but it also admits some infinite ones, for example, the monotone alignment on the positive in-

142

JAMI SON-HALDNER

tegers.

It would be interesting to know all spaces for which the conclusion

of (6.6) holds. 7.

CHARACTERIZING VARIETIES BY FORBIDDEN SUBSTRUCTURES As noted in Section 3, any variety is completely determined by its mini-

mal forbidden spaces (MFS).

For example, a space satisfies separation axiom

s1 iff it contains neither of the quasicircuits 00(2) and 01(2).

Also, a

space has rank < n provided it has no subspace isomorphic to F(n + 1).

(7.1)

The MFS for the variety of matrnids are the quasicircuits Qk(n)

for

0 < k < n and n arbitrary. (7.2)

There are seven HFS for (To):

namely, 00(2), Q1(Z), (20(3). 01(3),

(13(3), 17(2) x F(2), and the "Arrow" space [32].

The "Arrow" space is the subspace of four white nodes of the chordal graph in Fig. 3 under the monophonic alignment. Certainly the success of and interest in a characterization by forbidden subspaces depends on the number and complexity of the MFS. amples cited above, three have only finitely many MFS.

0f the four ex-

For such varieties,

the MFS characterization may be regarded as most effective.

We shall say

that a variety is finitely based if it has only a finite number of HFS.

Re—

call from (6.2) that this is equivalent to being definable by a single first FIGURE 3

THE ARROW SPACE

A PERSEECTIVE 0N ABSTRACT CONVEXITV order axiom.

143

The following is a list of some of the known finitely based

varieties:

(so)

'

(sl) (monotone )k

for all k

(rank < n)

for all 11

(rank < 1)k

for all k

(T0) (rank < n)

for all n

(53) f‘ (rank < n)

(PO) n

for all n

(matroid) n (rank < n)

(ED) 0 (rank < n)

for all 11

for all n

From the last four examples, one might be tempted to suspect that,

each :1, every variety contained in (rank < n) is finitely based.

for

This is

indeed true for n = 1 although the proof is not as trivial as one might eurpect.

For n > 3, the conjecture is false, however.

As Allan Day has ob-

served [10], the projective planes over finite fields with different charac—

teristics form an antichsin of rank 3 alignments in FinALN.

Hence the va—

riety of all alignments of rank < 3 which contain none of these planes as subspaces is not finitely based.

The problem remains open for rank < 2.

Even if a variety is not finitely based, its list of M'ES may not be all that bad.

described.

For example, as seen in (7.1) the MFS for matroids are easily

For order alignments (see [32]), they are more complicated but

still within reason.

For (Carath < n), the 1435 are awful:

one essentially

needs to describe every space with Carathécdory number n — l in order to describe the UPS of (Carath < n). circuits Qk(n) with k < n:

For free alignments, the MIPS are the quasi—

not so bad, but certainly not as simple as the

free alignments themselves. As a means to measure this complexity of a variety, this author would like to propose the following construction.

-

For any variety V, let (MES (V))

be the variety generated by the minimal forbidden subspaces of V.

Now let

the core of V be the subvariety generated by the infinite spaces in (MPS(V) ).

From the Upward Lfiwenheim-Skolem Theorem (page 67 of [5]), one can show that

144

JAMISON-HALDNER The finite spaces in core(V) are precisely those which lie in infi-

(7.3)

nitely many MFS of V.

From this,

it follows that

For any variety V, core(V) E V.

(7.4)

Also core(V) 'f ‘1’ iff V is not fi-

nitely based.

Thus the size of core(V) in V measures the effectiveness of HFS characterization for V.

From (7.1), we have corehnatroids) = (free), only a very

small portion of mattoids.

0n the other hand, core(free) = (free), indi-

cating the ineffectiveness of M35 classification in this case.

For (Po),

the core contains only spaces arising from orders of height 3 or less, again

a small portion of (P0). In contractible varieties, one can simplify the characterizations by considering minimal excluded minors (recall Section SB) instead of forbidden subspaces.

In the variety of matroids, this has become the accepted method

of characterization. trivial examples.)

(See Chapter 6 of [58] for a discussion and some nonThe examples below can be derived at once from the defi—

nitions.

(7.5)

An aligned space is a matroid iff it has no minor isomorphic to (21(2).

(7.6)

An aligned space is an antimatroid iff it has no minor isomorphic to

Q00).

(7.7)

An aligned space is free iff it has no minor isomorphic to either

(20(2) or 010% (7.8)

The minimal excluded minors for (Carath < n) all have n + 2 points--

hence there are only finitely many such excluded minors.

8.

SOME OPEN PROBLEMS Perhaps the most intriguing problem in the theory of varieties is the

characterization of those finite aligned spaces which can occur as subspaces of Euclidean space with ordinary convexity.

the question asked is this:

In the language of varieties,

A PERSPECTIVE 0" ABSTRACT CONVEXITV PROBLEM I.

145

Characterize the variety (Rd) generated by ordinary convexity on

Rd. As noted in Section 5]), the varieties (Rd) all lie in (T0)°°.

(Refer also

to’ R. Pollack's chapter in these Proceedings.)

PROBLEM 2.

Find an infinite class of [CPS for (Rd) which lie in no)" n

(Carath < d + 1). - Additional problems are noted belaw.

PROBLEM 3.

Describe the lattice of produce—closed varieties.

PROBLEM 4.

Do all prnduct—closed varieties arise from "separation axioms"

of some kind? PROBLEM 5.

(Sections 46, SC, and 6A)

Under the operation V - H the class of varieties forms a com—

mutative semigroup. PROBLEM 6. rieties.

(Section SC)

What is the structure of this senigroup?

(Section 5D)

Give syntactic descriptions of contractible and join-closed va— (Sections SB, 5D, and 6A)

PROBLEM 7.

Characterize the variety (T0)m.

(Section 51)).

PROBLEM 8.

Is the variety n finitely based for each 11?

(Sections SD

and 7) PROBLEM 9.

based?

More generally, if V and W are finitely based, is V - H finitely

That is, in reference to Problem 5, do the finitely based varieties

form a subs-igroup?

(Sections 5D and 7)

PROBLEM 10.

Are all rank 2 varieties finitely based?

PROBLEM 11.

Are all lattice-closed varieties finitely based?

PROBLEM 12.

What _is the structure of FinALN as an ordered set?

3)

(Section

146

JAMISON-NALDNER

ADDENDUM Jeff Kahn and J. P. S. Kung (AMS Bull. 3(1980), 857) have announced a classification of all varieties of matroids.

Their notion of variety assumes

contractible and several other properties not discussed here.

A. J. Hoffman

has kindly reminded me of Richard Rado's classic paper (J. London Math. Soc. 22(19A7), 219—226) in which the connection between Helly's theorem and the Chinese Remainder Theorem is noted and investigated (cf. Section 2.A).

Paul

E. Edelman (Alg. Universal—is 10(1980), 290—299) has independently shown that

the variety of antimstroids is closed under joins (cf. Table I, Section 5) and that they are characterized as dually locally distributive lattices (cf.

Table II, Section 6).

A recent book by G. Gierz at aZ.. (A Cowandium of Con—

tinuous lattices, Springer Verlag, Berlin,

1980) has considerable impact on

the theory of alignments since alignments are a special class of continuous lattices.

In fact, as J. D. Lawson has noted (private communication), con-

tinuous lattices may be viewed as "fuzzy" alignments.

ACKNOWLEDGMENTS The author is grateful to numerous people for their assistance and kind encourag-ent with this report.

Space does not permit thanking them all indi-

vidually here. but special acknowledgments are surely due to Rudolf Willa and the faculty and guests of Arbeitsgruppe I in Darmstadt for many valuable

conversations, especially during the author's visit in 1979; to Stefan Foldes for pointing out the syntactic approach to varieties; to Hal Kierstead for as—

sisting with an understanding of first order logic; and to the conference organizers, David Kay and Marilyn Breen, and to wife Dori, for their patience

and encouragement during the writing of this report.

The author is also grate-

ful to the Alexander von Humboldt Stiftung for the opportunity of spending 1976—1977 at the University of Erlangen where this work was begun.

This report was prepared with partial support from NSF Grant RES—8001553.

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J. R. Reay, Caratheodory theorems in convex product structures, Poo. J. Math. 35(1970), 227-256.

1:9.

H. E.

trand, Princeton,

1965.

Scarf, An observation on the structure of production sets with

indivisibilities. Proc. Natl. Acad. Sci. USA 710(1977), 3637-3641. 50.

J.

Schmidt, Ueber die Rolle der transfiniten Schlussweisen in einer

allgemeinen Ideal-theorie, Mth. Nachr. 7(1952), 165-182. 51.

J.

Schmidt, Einige grundlegende Eegriffe und Ssetze sus der Theorie der

Huellen-operatoren, Bericht usher die mthematiker—Tagung in Berlin, Januar 1953, Deutscher Verlsg der Wissenschaften, Berlin, 1953, 21—48. 52.

G. Sierksma, Caratheodory and Kelly numbers of convex product structures, P110. J. Math. 61(1975), 275-282.

53.

G. Sierksma, and Reay, J. R., A Tverberg-type generalization of the Kelly number of a convexity space, Preprint, Western Vashington Univ.. 1979.

54.

A. Tarski, Fundamentale Begriffe der Methodologie der deduktiven Wissen— schaften, Monatsh. Math. Phys. 370930), 360404.

55.

N. T. Trotter, and Moore, J. 1., Characterization problems for graphs, partially ordered sets, lattices, and families of sets, Discrete Math.

. 16(1976) , 361—381. 56.

H. T. Tutte, A homotopy theorem for matroids I, II, Trans. AMS 88(1958), 144-174.

57.

H. T. Tutte, Man—aids and graphs, Trans. AMS 90(1959), 527-552.

58.

H. T. Tutte. Introduction to the Theory of Matroide, American Elsevier,

New York, 1971.

150

JAMISON—HALDNER

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H. Tverherg, On equal unions of sets, Studies in Me Math. , L. Mirski, ed., Academic Press, London (1971), 249—250.

60.

M. van de Vel, Pseudo—boundaries and pseudo-interiors for topological convexities I, II, Rapport M». 100, Vrjie Universiteit Amsterdam, 1979.

61.

H. van Masren, Pseudo—ordered fields, Indag. Math. 36(1974), h63-h76.

62.

H. van Maaren, Algebraic Simplices in Modules: 0n Concepts of Gloom and Generalizations af' Convenity, Dissertation, Rijksuniversiteit Utrecht, 1977.

63.

D. J. A. Welsh, Matroid Theory, Academic Press, London, 1976.

64.

H. Whitney, On the abstract properties of linear dependence, Amer. J. Math. 57(1935), 507-533.

65.

R. Wille, Kongruenaklassengeometrien, Lecture Notes in Math. #113,

66.

R. M. Wilson, An existence theory for pairwise balanced designs, I, J.

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0. Zariski, and Samuel, P., Commtativs Algebra, 1, Van Nostrand, Prince— ton, 1958.

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J. Kahn, and Kung, J., Bull. MS, 30980), 857-858.

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Comb. Th.

(A) 13(1972), 220-245.

NOTE BY EDITOR.

Problem 1 mentioned in this paper has been partially

solved, although various solutions were known previously in unpublished manuscripts.

A little known paper by J. P. Doignon [Caractérisations d'espaces

de Pssch-Pesna. Acad. Roy. Belg. CZ. Sci.

(5) 62(1976), 679—699] deals with

the most difficult aspect of embedding an alignment of dimension > 2 with certain properties

vector space.

(A, B, C we shall call them)

'A paper by J. H. H.

in a convex subset of a real

Whitfield and S. Yang [A characteriza-

tion of line spaces, (Jan. Math. 31411., 24 No. 30981), 351—357] then shows that an alignment satisfying very natural convexity properties of Ru (denoted JHC, REG, STR and (114?, shown to be independent in the paper) also possesses the properties A, B and C and thus, by the previous result, is isomorphic

to an aligned subspace of a real vector space.

Doignon's theorem was es-

tablished for dimension > 3 independently by J. Cantwell and this editor [Geometric convexity.

III:

Embedding, Trans. AMS 256(1978), 211-230] using

conventional methods in the foundations of geometry; Doignon's argument makes use of a 1938 theorem of Spemer.

OPEN PROBLEMS AROUND RADDN'S THEOREM John R. Reay

_ ’

of '

hr“

Les mid f‘

em‘ w.

r

western Washington University Bellingham, Washington

1.

INTRODUCTION The well-known theorem of Radon [1921] asserts that

THEOREM 1.1

Each set of d + 2 points in Rd is the union of two disjoint

subsets whose convex hulls have a common point. The purpose of this lecture is to list a number of open problems and conjectures that are related to this classical theorem of Radon; and iden—

tify some of the recent work that generated or partially solved these problems.

No proofs will be given, and no attempt is made at making this a com—

plete survey.

For good surveys of the area and its bibliography, see the

essays of Danzer—Grfinbaum-Klee [1963] (for early literature), Doignon-Valette [1975], and Eckhoff [1979], and also the expository article of Peterson

[1972]. Much of the work of the last decade related to Radon's Theorem has been

in an abstract setting, but we deliberately leave this until last since (a) the familiar Rd is certainly the best place to begin and is the source model for most axiomatic treatments, (b) Gerard Sierksma will cover the most

151

REAY

152

recent abstract results in the next lecture, and (c) the open problems and results in Rd should always be kept in mind when considering abstract con—vexity.

We will first consider independence and (m,k)—divisib1e sets, them primitive Radon partitions and convex polytopes, and finally the alignment (abstract convexity) setting .

2.

INDEPENDENCE IN (m,k)-DIVISIBLE SETS

A set 5 c Rd is (m,k)-dimlsible if it may be partitioned into In (pairwise disjoint) subsets whose convex hulls intersect in a set of dimension

at least 1:.

Such a partitioning of S is called an (mm-partition.

empty set has dimension —1; assume 2 < In and 0 < k < d.)

set is also called m—divisible. (d + 2)-set in Rd is 2—divisible.

(The

An (m,O)—divisib1e

Thus Radon‘s Theorem asserts that each Birch [1959] conjectured and Tverberg

{1966] proved the following THEOREM 2.1

Each [(m — 1) (d + 1) + 1]-set in Rd is Ill-divisible.

Although Tverberg's proof made use of algebraic independence of sets (i.e., the t points of 5 have t-d algebraically independent real coordinates

over the field of rationals), his proof can be modified to use the weaker condition of strong independence which essentially asserts that the points of 5 do not form pencils of lines, or books of planes, etc.

Set S C Rd is

strongly independent provided each finite collection 51,...,S;: of disjoint

subsets has the property:

if card 51 = di + 1 < d + 1 then dim(n:=1 off 51)

= max(-1, d — {i=1 (d — di)}. That is, for any appropriate collection of flats, the codimension of the intersection is the sum of the codimensions. Thus strong independence of a set S clearly implies general position. For a set to be (m,k)—divisib1e with k > 2 some independence is neces— sary to avoid having all the points on a line.

It is an open problem to

determine this best (weakest) independence.

PROBLEM l.

Conjecture:

Each [(m - l) (d + l) + k + l]-set in general posi-

tion in kcl is (m,k)—divisib1e. Peterson [1972] gives a geometric proof in case m = 2; Reay [1968] proves

the case d = 2; and Reay [1979] establishes the case :1 = m = 3.

If the

OPEN PROBLEMS AROUND RADON'S THEOREM

153

FIGURE 1

hypothesis of general position is strengthened to strong independence then the conjecture is true as a corollary of the following result by Reay [1968]:

, THEOREM 2.2

If the [(m - l)(d + l) + k + l]-set S C Rd is strongly inde-

pendent and S = S

U --- U SIn is a partition with each card S1 < d + 1, then 1 if Fq_1 conv S1 is nonempty, it is of dimension k.

The example shown in Fig. l, where d = k = 2, m = 3, consists of nine points in general postion which are (3,2)—divisib1e even though the indicat-

ed partition shows that 2.2 fails.

Thus, strong independence cannot be re-

placed by general position. A cheap generalization of Radon's Theorem is obtained by replacing the field of real numbers by any ordered division ring, since the usual proof (Radon's original) remains valid.

But Tverherg's proof of 2.1 relies in

several places on properties of the real numbers and does not have such a trivial generalization.

Doignon and Valette [1977] show that 2.1 does re-

main valid in any d—dimeneional affine space Ad over an ordered division

ring (i.e., skew—field), and give a more general proof of Tverberg's Theorem. It turns out that the natural independence condition to consider in this setting is fall independence, which is defined in the same way as strong independence except that aff Si is replaced by proj 51’ the projective subspace generated hy Si in the projective extension of Ad.

Surprisingly,

strong independence implies full independence only in spaces of dimension less than five, while the four_vertices of a parallelogram in any affine space show that full independence does not imply strong independence. ’fact, Doignon and Valette [1977] show that there exist finite strongly

In

154

REAY

independent sets which are maximal under inclusion, and characterize them

as the sets which are strongly independent but not fully independent! also Doignon—Valette [1975].)

Eckhoff

(See

[1979] remarks that 2.2 remains valid

with full instead of strong independence.

PROBLEM 2.

Is there a better general setting in which both full and strong

independence become extensions of a more general (weaker) independence? Next note that no independence at all is required in the special case

k = 1, that is, when we wish to partition a set s into subsets whose convex hulls have a common line segment.

However, only crude upper bounds are known,

in general, on the cardinality of S necessary for

(m,1)-divisihility.

Reay

[1979-b] shows that each (2(d + 1)(m - 1) + l)—set in Rd is (m,1)-divisib1e. A much more reasonable conjecture about the correct cardinality is the fol— lowing.

PROBLEM 3.

Conjecture:

Each (2d(m - 1) + 2)-set in Rd has a ((d + 1)(m - l)

+ 2)-subset which is (m,l)-divisible.

Easy examples show that the number 2d(m - 1) + 2 cannot be reduced;

it is clearly true if d = 1, and it was proved for d I 2 by Reay [1979—1)] and for m = 2 by Eckhoff [1976].

A related question of Doignon which re—

quires no independence condition is

S = S

l

Conjecture: Each (2d + l)-set s C Rd has a Radon partition ' ' U 52 such that d1m(conv S1 n conv 52) > min(dim Sl’ dim 52}.

Attempts to establish the conjecture in Problem 3 produced the follow—

ing notation which has been useful in abstract convexity. Point x is an

shin .

PROBLEM lo.

m—divisible point of set S C Rd if x E nil-1 conv S1 for some partition 5 51 U --- U SIll of S.

We denote the set of m—divisible points of S by Dm(S).

The next result seems to support the conjecture in Problem 3 (see Reay [1979—11]! THEOREM 2.3

Each (2d(m — l) + 2)—set in Rd has at least two distinct m-

divisible points.

'

Also let cm(s), called the m—care of S, denote the set of all points y E Rd

OPEN PROBLEMS AROUND RADON'S THEOREM

155

so that each closed half—space containing y also contains at least an points of S.

The m—core of S may equivalently be defined in a way which generalizes

to on abstract convexity space:

cm(s) = New; T | T c s and card(S ~ T) < m — 1} It would certainly be of interest to know under what conditions the set Dm(5)

of m—divisible points is a convex set, since‘2.3 then implies that the con— jecture in Problem 3 is true.

PROBLEM 5. (b)

(a) when is Dm(S) convex? Conjecture:

If S C R“1 is finite, so Cm(S) is a polytope, then

the vertices of Cm(S) lie in Dm(S), or equivalently, Cm(S) =

conv(Dm(S)). (c)

Under'what conditions is core Cm(S) at least k-dimensional?

It is clear that Dm(S) C Cm(S), and the methods of Birch [1959], ap— plied to planer sets S, show that Dm(S) = Cm(S) whenever 3m < card S.

Fur-

ther results and open problems concerning the m—core Cm(S) will be given by Sierksms. A possible first step towards solving Problem 2 and unifying some of the above results has been made by Doignon [1980]. that the general position of S

His first result shows

(any subset of at most d + 1 points is af—

finely independent) is too strong a condition in Problem 1 in case 11: - 2

and k > 2.

(The example + in R2 shows why k > 2 is necessary.)

THFDREM 2.3A.

(3)

A (d + k + 2)-set S C Rd is (2,k)-divisible for 2 < k

< :1 provided each d + 2 points of S sffinely span Rd. that each d + 1 points affinely span Rd.)

(b)

(It is not necessary

More generally S C Rd is

(LU-divisible if card 5 > d + k + 2, k > 2 and S "' T affinely spans Rd whenever 2 < card ’1‘ < k.

(Combining this result and the known cases of Prob—r

lem 1 gives the next even more general result.)

(c)

A set 5 C Rd of at

least a(k,d) - (d - k + 2)k + 2 points is (2,k)—divisible provided no 2k 4* 2 of them lie in a k-dimensional flat, end 2 < k.

PROBLM 6.

(a)

Can the bounds u(k,d) given in 2.311 be lowered?

REAY

156

The bound u(k,d‘) = (d - k + 2)}: + 2 is best if k = 2 or d - l or d, but Problem 6 is open otherwise.

We say a set is k-indepsndent if any k + 1 or

fewer of its points are affinely independent.

Thus k-independence is weaker

than general position unless k = d, in which case they are equivalent.

Eck-

hoff has commented that 2.3A implies the following:

THEOREM 2.4. A set s c Rd of at least B(k,d) - (d — k + 2) (k + 1) points is (2,k)-divisib1e if it is k-independent. Again the bound on card S is best if k = 0 or 1 or d.

PROBLEM 6.

(1))

Can the bounds B(k,d) given in 2.4 be lowered?

It would be nice to know how the above two results can be extended to (m,k)-divisible sets; Problem 6 might be the key to that project.

PROBLEM 7.

Conjecture:

Any k—independent set S C Rd of at least (m - l)

(2d — k + 1) + k + 1 points is

(m,k)-divisible (k > 0).

-

(In fact, any such

S would have a [(m - l) (d + l) + k + l]—subset which is (m,k)-divisib1e.)

Note that if k=d this reduces to a case of the conjecture in Problem 1, while if k - 1 so that l-independence just implies distinct points, this becomes Problem 3.

The standard example, which shows that the. number 2d -

(m - 1) + 2 in Problem 3 cannot be reduced. is s - {uh I b e 3, o. - o, :1, ..., t(m - 1)} where B is any linear basis for Rd. (mu—divisible nor 2—independent.

This set S is neither

Thus it is tempting to conjecture a

stronger version of Problem 3 when S is 2-1ndependent.

PROBLEM 8.

A 2—independent

(2d(m — 1) + 1)-set in Rd is

(m,l)-d1visib1e.

It seems unlikely that the cardinality constraint in Problem 8 is the best possible. Tverberg [1968] extended 2.1 as follows

1mm 2.5. Let m> 2,0 < c < a, s = (m - Me 4 1) + 1. and let FIJI. ""Fs be flats of codimensiun c in Rd. Then {1,...,s} has an m-psrtition m A1, . . . 'Am so that n 1_1 (conv ujEAi F1) contains a flat of codimension c .

OPEN PROBLEMS AROUND RADON‘S THEOREM

157

This result reduces to 2.1 when c = d.

By choosing all the flats F1

to be parallel in Rd ‘when c < d it is easy to see from 2.1 that the bound on -S is the best possible.

But the effect of making the flats Fi independent

has been investigated only for the case where no algebraic condition exists.

See Tverberg [1968] and Iversland [1969, Chapter A].

PROBLEM 9.

(a)

"hat independence conditions on the flats 171 of 2.3 will

allow the codimension c of the intersection to be reduced? (1:)

By increasing the number of flats by k (l < k < c), what independ-

ence condition will assure the intersection in 2.5 is dimension at least d — c + H (c)

what results analogous to 2.5 are possible if the flats F1 are al—

lowed to have different dimensions?

Robert Jamison has suggested a different variant of 2.1 and (m,k)—

divisibility (S is (m,k)—divisible if there is a partition S = Sl U --- U Sm with If;

couv S 1 k-dimensional).

Suppose we only require conv S

l to be k-dimensional for each pair (1,1), 1 < i < j < m.

n conv S

i Perhaps a smaller

cardinality for S would suffice for this weaker condition.

PROBLEM 10-

(a)

Each set S C Rd of at least T(d,m,n,0) -

(m - 1)(d + l) + 1

points has an m—partition S = S1 U --- U Sm so that each n of the sets (conv Si I i - l,...,m) has a nonempty (at least 0-dimensional) intersection.

This is a weaker form of 2.1.

Conjecture:

If 2 < n < m, the number T(d,m,n,0) given above is the best

possible .

Clearly the case n - m is Just 2.1.

The conjecture states that the

cardinality condition of Tverberg's theorem cannot be improved in general when the intersection requirement is weakened (m > n).

The conjecture fol—

lows directly from Helly's Theorem if n > d + 1, and is therefore trivial if d - 1.

It is shown in Reay [1979-1)] that the conjecture is true if d - 2

or if d = m = 3 and lower bounds on T(d,m,n,0) are given. We may define T(d,m,n,k) similarly, where k denotes the minimal dimen. sion_of the intersection of any n of the 111 sets {conv Si).

additional independence condition on S is again necessary.

men k > 1 an

1 58

REAV

PROBLEM 10.

(b)

Investigate the various independence conditions in deter—

mining the minimal value of T(d,m,n,k).

(c) Conjecture: If s c Rd is k—independent and n > 2, then T(d,m,n,k) *1 (m — l) (d + 1) + k + l (and thus T is independent of u). If an (m,k)—divisible set has an abundance of points when forming an (m,k)—partition, then it would be reasonable to distribute the excess as

uniformly as possible among the m subsets of the partition. 'iust in case some of the points get "stolen" or "lost" in the future.

a

Let L(d,v) be the small—

est integer so that each set S C Rd of at least L(d,v) points has a (2,0)—

partition 5 = S1 U 52 with the property that Di

=1

conv(Si "' T) 3‘ 0 whenever

any subset T C S which is "stolen" has cardinality at most v.

PROBLEM 11.

(a)

Determine L(d,V)-

[L(d,0) = d + 2 is just Radon's Theo-

rem.)

(b)

Conjecture:

Ezample:

L(d.v) - (v + 1) (d + 1) + 1-

To see that the conjecture is true when d - 1, consider 2v + 3

points on a line alternately in 51 and 52.

This number is best, because any

1 1

partition 51 U 52 of 2v + 2 points would need exactly v + 1 in each 51' yet taking all v from one set leaving only the last point on the line in 51, say,

would make “i=1 conv(Si ~ T) empty. The hard part of the conjecture, namely L(d,v) < (v + 1) (d + 1) + 1, was proved by Larman [1972] in case v = l and by Strangeland (1978] for high-

er values of v.

So "all that is needed" are the counter examples of [(v + l)

(d + l)]—sets in Rd, no (LN—partition of which can afford to have v points stolen. It was natural to extend the definition of L(d,v) to

(m,k)-partitions.

Let L(d,m,k,v)

from (2,0)-partitions

denote the smallest integer so that

each k-independent set S C Rd of at least L(d,m,k,v) points has an (m,k)— partition with the property that dim(f‘l':-1 conv(S1 "' T)) > k whenever T C S is any subset of at most v points.

(Here k < d and m > 2.)

Clearly L(d,m,0,0)

is determined by Tverberg's Theorem 2.1 and L(d,2,0,v) is Larman's function given above.

PROBLEM 11.

The following conjecture is due to Strangeland [1978].

(c)

Determine L(d,m,k,v), or determine 1. for the more restric-

tive class of strongly independent sets in Rd.

(d)

Conjecture:

L(d,m,0,v) = (m - l)d(v + 1) + m + V-

1

OPEN PROBLEMS AROUND RADON'S THEOREM

-

159

See Eckhoff [1979] for related results and conjectures, some of which relate to the function T of Problem 10, and for the related problems con—

cerning a common transversal which meets each of the m (possibly disjoint) convex bulls formed by a partition of set S.

We state only the following

(Iversland [1969]).

PROBLEM 12. Um

i=1 51

Does every [(m - l)d + n + 1]—set S C Rd have a partition 5 =

so that among the convex hulls {conv,Si} there are n, each of which

intersect all m convex hulls? Let S be a (d + 2)—set 1n Rd, and let S = 51 U Sz he the partition guar— anteed by Radon's Theorem.

It is well—known that the partition is unique

if and only if S is in general position. metric proof.)

(See Peterson [1972] for a geo—

Further, this (2,0)-partition gives a unique single point

’ as the intersection of the convex hulls.

There is no known analogous re-

sult for (m,0)—partitions of a set S if m > 3. PROBLEM 13.

(a) For which sets 5 is it true that every (m,k)—partition of

5 gives the same (unique) k-dimensional polytope for the intersection of the corresponding convex hulls? (b)

which (m,k)-divisib1e sets 5 have a unique (m,k)—partitian7

If a set S has an (m,0)—psrtition then it is unique only if no proper subset has an (m,0)—partition. so we consider only such sets. Kmnple.

The set shown in Rs (Fig. 2) has a unique (3,0)-partition

{(0,0,2),(0,0,-2)} U {(0,2,0),(0,-2,0)} U {(-1,0,1),(-1,0,—1),(1,—1,0),(1,l,0)}Since C3 (5) - ((0,0,0)} = 133(5), the origin is the only 3-divisib1e point.

Yet this set of (n - 1) (d + 1) = 8 points is not in general position. the other hand, any

many (3,0)—partitions. PROBLEM 1A.

(a)

On

(3,0)—divisible set of at least 9 points appears to have

A lower bound has been conjectured by Sierksma.

(Dutch Cheese Problem)

Conjecture:

If S C Rd is an

[(m - l) (d + 1) + l]-set which is (m,0)—divisib1e, but no proper subset of

S is (m,O)-divisih1e. then S has at least [(m - 1)!]d distinct (m,0)-parti— tions. _(b)

' If S C R

d

‘ is a strongly independent [(m — l) (d + l) + k + l]—set,

'what is the minimum number of distinct

(m,k)-partitions?

REAV

160 FIGURE 2

The bound [(m — 1)!)d is based on the fact that d + 1 independent points, each of multiplicity m - 1, together with a single point interior to this simplex,

forms a [(m — l) (d + l) + l]-set of

number of distinct (m,0)—partitions.

(nondistinct) points with this

The name of the conjecture comes from

the feet that Sierksma [1979] offers a Dutch cheese for the first solution. If m I 2, so that the bound reduces to l, and card 5 - d + 2, the problem

reduces to the well—known result quoted above.

3.

CONVEX POLVTOPES AND PRIMITIVE RADON PARTITIONS We bridge the jump from the last section with this problem posed by

Tverberg in the 1978 Oberwolfach conference on convexity.

PROBLEM 15.

Conjecture:

polytope, and 1r:

If p c R(m—1)(d+1) is a (nondegenerate) convex

bd P ~> Rd is a continuous projection then there exist pair-— m

"I

vise disjoint faces (F1)i=i of P so that 01:1 «(1:1) 3‘ d’.

Since any [(m — l)(d + 1) + lI—set in Rd is «(m for some simplex 1‘ in ROD-1) (d+1), this conjecture would give an alternate proof of 2.1.

The

case n = 2 was proved by Bejméczy and Bérény [1979) (giving an alternate proof of Radon's Theorem).

More generally, the conjecture was provedjay

Bérény, Shlosman, and Szllcs for all cases where m is a prime!

OPEN PROBLEMS AROUND RADON'S THEOREM

161

The following problem was posed by Peter McMullen.

PROBLEM 16.

Is Zd + 1 the_ maximum number of points in general position in

Rd (but otherwise arbitrarily placed) which can be mapped, by a permissible

projective transformation, onto the vertices of a polytope in Rd? At first glance this problem has little to do with Radon's Theorem. But by use of Gale diagrams it is a reformulation of one case of Problem

110:) above.

(See Larman [1972].)

In fact, Gale diagrams have played a role

in a number of problems around Radon's Theorem.

For example, Shephard [1969]

has characterized the unique Radon partitions {S 1, 52} of a (d + 2)-set S in general position in Rd by the fact that the Gale transforms E1 and E2 lie on opposite sides of the origin in the transform space R 3).

(b)

The (m,k)-partitions of S when S has another independence, such as k-independence or strong independence (m > 2).

Most of the remaining results and problems of this section are stated for (2,0)—partitions, called Radon partitions, of sets S.

However, they

generally could be‘rephrased in the setting of (nun—partitions. Si of a Radon partition 5 - S 1:

The sets

U 52 are called components of the partition.

{T1,T2} is a Radon partition in s (as opposed to of 5) 1f 1: is a Radon partition and if T1 U T2 is a subset of S.

The partition {51,Sz}ea:tefldas {T1,T2} .

REAV

162 if Ti C Si"

{T1,T2} is a primitive Radon partition in S if it is minimal in

the sense that it does not extend any distinct Radon partition in S.

M

sets 5 and T are said to be Radon equivalent provided there exists a bidettion f : S + T which preserves Radon partitions, or equivalently a bijection g :

S -> T which preserves primitive Radon partitions.

The equivalence ,classes

induced by this equivalence relation are called Radon types. [1974, 1975, 1979].

See Eckhoff

These definitions will be thought of in terms of sets

5, T C Rd, but these terms along with simplex, general position, faces, 1(-

neighborly, and most of the following results and problems can be formulated for sets S and T in any real linear space (finite or infinite dimensional).

See Petty [1975].

We collect some well—known facts about primitive parti-

tions in: THEOREM 3.1. (1:)

(a)

Each Radon partition extends a primitive one.

For each primitive partition {A,E} in S

(l)

A U B is in general position in aff(A U B).

(2)

card(A U B) = dim(A U B) + Z, is finite, and is < d + 2 if S C Rd.

(3)

A U B is (2,k)-divisible iff k - O, and its Radon partition is unique.

(1.)

points x,y E A U 3 lie in different components iff aff(A U 3 ~ {x,y}) separates x and y in aff(A U B).

-

(c) If s C Rd has d + 2 general position points, not all on a common (:1

l)—sphere, and S = A U B is its unique primitive partition, then

A = {x E S

I

x lies inside the

(d - l)-sphere through S ~ x)

and

n = {x e s | x lies outside the (d — l)-sphere through 5 ~ x} Primitive Radon partitions were first used by Hare and Kennelly [1971] in an attempt to discover when certain points of a set S lay in the sane com-

ponent of s Radon partition.

They obtained part (c) of 3.1 above and the

following for the special case when S is a finite set in general position and in Rd.

The present form is due independently to Petty [1975] and Doignon—

Valette [1975]. THEOREM 3.2. if and only if

'

A subset T C S lies in a component of some Radon partition of 5

J

OPEN PROBLEMS AROUND RADON'S THEOREM

163

(a)

3 ~ T is affinely dependent, or

(b)

conv T n afif($ "' T) f t?

PROBLEM 18.

(a)

when does a subset T C S lie in one (or in all but one)

component of a (m,0)—partition of 37 (b)

(m > 3)

when does a subset T C S necessarily meet every (or at least two)

component of a (m,0)—partit10n of S?

(In > 2)

In general, it seems to be very difficult to say when two finite sets

in R‘1 are Radon equivalent. PROBLEM 19.

Characterize the Radon types of finite sets of points in Rd.

Progress has been made only for certain types of sets.

Breen [1973]

' implicitly characterized the Radon types for the vertex sets of cyclic polytopes.

Eckhoff [1975] used Gale diagrams to classify the Radon types of

general position sets with at’ most d + 3 points.

Any two such sets, which

are vertex sets of d-polytopes, are Radon equivalent if and only if the polytopes have the same combinatorial type.

More generally, Breen [1972] showed

that the combinatorial type of any polytope is determined by the Radon par—

titions of the vertex set, and in fact, by the primitive Radon partitions which make use of a (fixed but arbitrarily chosen) vertex v.

However, easy

examples show that combinatorial equivalence of polytopes does not imply Radon equivalence of their vertex sets.

The key result of Ereen was the

following characterization of the facial structure of d-polytopes.

THEOREM 3.3

Let S be the vertex set of a d-polytope and F C S.

The set

conv F is a face of conv S if and only if T1 C F " '1'2 C F for every primitive Radon partition {T1, T2} in S.

In particular F determines a face of a

simplicial polytope eonv S if and only if no subset of Y is a component of a

Radon partition of S.

PROBLEM 20. tions in 5

(a)

Given a finite set S and a minimal collection C of parti-

(minimal in the sense that no partition in the collection extends

another), when does there exist a polytope in Rd whose vertices correspond

‘to s and whose primitive Radon partitions correspond to the partitions of C7 (b)

If such a palytope exists, what can be said about the range of d?

164

REAY See Bland and Les Vergnss [1978] for a statement of this problem in the

setting of orientable matroids. There have been a number of results about the number MS) of distinct Radon partitions that a set S C Rd may have.

See Eckhoff [1979].

One of

the most appealing is the following due to Hinder [1966].

THEOREM 3.4.

Ms) is the number of subsets T C S for which dim T + card T

is even. PROBLEM 21.

what is a "natural" correspondence between the Radon partitions

of S and the subsets T C S with dim 1‘ + card T even?

Theorem 3.5 was not stated by Winder in the form given above; in fact,

Vinder never mentions Radon's Theorem at all. work lies in a definition of Brylawski

The connection with winder's

[1976], who lets R(S) be the number

of subsets of S C Rd which can be separated from their complements by a hyperplane.

Radon's Theorem then becomes:

R(s) < 2‘1 if card S > d + 2.

There are a number of results which give an upper bound on “(5) for various classes of sets S.

(See Brylawski [1976, 1977].)

For lower bounds

it is natural to consider primitive partitions of S.

PROBLEM 22.

(a)

Deter-mine the smallest number of primitive Radon partitions

of a set of s points in Rd. (1:)

For which set is the lower bound attained?

Determine bounds on the number of distinct

(m,k)-partitions of a

set S C Rd in terms of card 8 and the various independence conditions that may be imposed on S.

This problem should be compared with the Dutch Cheese Problem (Problem 14).

See Lindquist-Sierksma [1980] and Sierksma [1980] for related results

in an abstract convexity structure.

See Doignon [1980b] for a setting of

the problem using matroids, a general conjecture, and proofs of certain special cases. It has long been an open problem to determine which vectors 1' - (fO'fl' "“fd—l) of natural numbers represent the f—vectors of a d-polytope if d > A.

A similar problem arises when we study the sizes

in S is of type {tl,t2) if card T1 = :1.

A partition ”1’12}

For a set S C Rd of 5 points, let

I‘M

(See Barnette-Resy [1973].)

of the components in Radon partitions of a set S C Rd.

OPEN PROBLEMS AROUND RADON'S THEOREM

165

r:i I r1(S) be the number of Radon partitions of S of type {1,s — i) and let p1 = pi(S) be the number of primitive partitions in S of type {1,d + 2 - i).

Then the vectors (r1....,r[E/2]) and (p1,...,p[d/2]+1) are called the Radon vector and the primitive Radon vector of 5, respectively. PROBLEM 23.

(:1)

Which vectors (r1, . . . ,r[E/2]) and (p1,. . . 'p[d/2]+l) of

natural numbers represent the Radon vector and primitive Radon vector,

re—

spectively, of some set S of s points in general position in Rd? (b)

Find upper or lower bounds on r1 or pi for sets of s general posi-

tion points in Rd.

Except for certain cases this appears to be a very hard problem.

The

analogue of the Euler hyperplane for f-vectors are the hyperplanes

[d/2]+1

i=1

5

_

Pi ' [an]

(3‘4)

and

{i312} r1 = NS)

(3.5)

Equation (3.1») is the only linear relation on the coordinates of every primitive Radon vector, but other linear relations than (3.5) hold for the Radon VECCOI’B-

If s - d + 2, the uniqueness of the (primitive) Radon partition of 5 makes the characterization trivial; all coordinates are zero except for one whose value of one could appear in any position. If s > d + 3, then the Radon vector and the primitive Radon vector of S will have an initial segment of zeros as coordinates and thereafter will al—

ways be positive.

Furthermore, the vertices of a k—neighborly polytope which

is not (k + 1)—neighbor1y will have a vector whose initial segment of zeros is

of length exactly k.

This was proved for Radon vectors by Shepherd [1969]

and for primitive Radon vectors by Eckhoff [1974] using Gale transforms. when s - d + 3,

so that the Gale diagrams are two-dimensional, Eckhoff also

gave a complete characterization of'the Radon and primitive Radon vectors of S, and showed that either vector determined the other. If s = d + 10,, then knowing either the Radon vector or the primitive

Radon vector of a set S does not allow us to determine the other vector. But here the two vectors are related through the f-vector of conv S, as was shown by Kramer [1975]:

REAY

166 If S C Rd is in general position,

card 5 = s = d + In, and f is

then

1 5-3' fj_1 r1 —_ p1_1 — [1-0 (-1) 1'j [1-3]

when i —_ l,...,[d/2] + 1

Theorem 3.6 will not hold if 3 becomes large compared to d in general.

PROBLEM 24.

Characterize the sets S of at least d + 3 points for which 3.6

is true, i.e., primitive Radon vectors are determined by the Radon vector of

S and the f-vector of conv 3.

Note that if Problem 24 were established, then the Dehn—Soumlerville

.r.. _« _ .._....n.e_ .__~.

the f-vector of conv S,

..a

THEOREM 3.6.

equations for conv S would yield another linear relation between the ri and Pi. PROBLEM 25.

Find a complete system of linear relations which are satisfied

Several such linear relations have been found by Kramer [1975].

The

necessary calculations make fundamental use of Breen's [1973] characteriza-

tion of the primitive Radon partitions of the cyclic polytope. New proofs of Radon's Theorem continue to be found by investigators of other areas of mathematics.

For example, Dudley [1979] supplies a proof

based on a series of papers with probabilistic implications.

We close this

section with the following probabilistic problem:

PROBLEM 26.

Let S be a set of d + 2 points Chosen randomly in a bounded

open domain D C Rd.

Find the probability Pi that the unique primitive Radon

partition of S is of type T1 = {1, d + 2 - i) for i - 1,2,...,[d/2] + l.

The probability vector (P1,...,P[d/2]+l) obviously depends on the domain D, but this restriction could easily be replaced by requiring only that the origin is one of the d + 2 points of S, and the other points are chosen randomly in Rd and form a set of diameter at most one. considered in the plane long before Radon's time.

This problem was

'Sylvester

(while study-

ing calculus of variations) conjectured that a disk (the interior of a circle) was the domain D C R2 which minimized the probability of four randomly chosen

.-_..a saw-nee. . v

by the Radon vector and the primitive Radon vector.

OPEN PROBLEMS AROUND RADON'S THEOREM

167

points from D having a Radon partition of type {1,3} (that is, one point lies

in the convex hull of the other three).

In a survey article written two

years before Radon was born, Grafton [1885] shows this probability is P1 = 35/(1211’2) when I) C R2 is a disk.

See Klee [1969] for a survey of more re-

cent references and results.

4.

ABSTRACT CONVEXITV SPACES The early survey paper of Danzer-Crunbamn—Klee [1963] listed a number

of ways in which the notion of convexity had been generalized.

Radon's The-

orem had been considered in many of these settings, but mostly as only one part of an investigation of a particular space or axiomatic structure.

One

suggestion of their paper was to "study the inter—relationships of Radon's, Helly's, and Carathéodory's properties in a system which had convex sets closed under intersection, and perhaps other axioms of convexity." tions are given below.)

(Defini—

A number of subsequent papers investigated this

interrelationship in a specific special setting such as convex—product spaces,

order convexities, real linear spaces, semilattices and lattices, spherical convexities, or trees, etc.

More recently this interrelationship has been

investigated in as general a setting as the definitions will allow, in the hope of getting results which are valid in most of the special settings. This latter effort was started by Kay and Womble [1971] and Jamison [1970] in particular, and carried on by Doignon—Valette [1975], Doignon-Reay—Sierksma [1979], R. Hammer [1977], Jamison [1979] ,' Sierksma [1977], and others. interesting side effect has been to turn the problem atom-Ad.

One

Instead of in—

cluding Radon numbers, etc. in a study of some particular structure, theorems about Radon numbers, etc. are being proved in a general setting, and then examples of particular structures are being sought in an attempt to prove

that the theorems are as sharp as possible. We now turn to some definitions.

The one common axiom is property (I):

the intersection of any collection of convex sets is a convex set.

An ab-

stmt convexity space is any pair (X,C) where X is a set and C a collection

of subsets (called convex sets) which satisfy o E C, X E C and property (I).

An alignment is an abstract convexity space with C closed under nested un— ions.

The C-hull (generalized convex hull) of a set S C X is the set C(S) =

“{A E C I S C A}.

The Ill-Radon number 1-011) of (X,C) is the least positive

» integer so that each set S C X has a Radon m-partition provided card S > r(m).

The number r = r(2) is called the Radon number.

The n-KeZZy number h(n) of

REAYV

168

(X,C) is the least nonnegative integer so that each set S C X has Cn(S)' # 0

provided card S > h(n) + l. ‘ (See Section 2 above for a definition of Cn(s), the n-core of S.)

The number h = h(2) is called the [Jelly number because

h(2) may equivalently be defined as the least nonnegative number so that the intersection of any finite collection of convex sets is nonempty provided the intersection of each subcollection of at most h(Z) of its sets is non— empty.

The exchange'number e of (X,C) is the least positive integer so that

for each x e x and each finite s c x, C(S) C U 6 S C(x U (5 ~ y)) Provided card 5 > e.

The Cwatheodory number a of (X,C) is the least nonnegative

integer so that C(S) = U{C(T)

I T C S, card T < c} for all S C X.

We summarize various known results in the following theorem: THEOREM 4.1. (a)

h(n) < r(n + l) - l

(b)

r k + 1 has the property that "(C(S \ (ad) I x E S} f 1’

Instead of a finite collection in ‘(a) one can take a collection of k + 1

sets, and in (b) it suffices to take [SI = k + 1; see Sierkama [23]. Theorem 8.1. If

Note that,

(X,C)

in general, 0 < h 1 the 1-Rad0n

number r(1) of an aligned space (X,C) is the infimum of all positive integers k with the property that:

Each set S in x with IS] > k admits a 1-partition

S = _S1 U ~-- U S.r into pairwise disjoint sets Si such that C(51) n

a! o.

Such a T-partition of S is called a Radon 1-partition of S.

(1 C(51)

Tverberg's

theorem states that r(r) = ('r - 1) (d + 1) + l for the ordinary aligned space

(Rd,conv).

To give a similar generalization of the Belly number we first

introduce the o-core of a set S in X (a > 0), namely

coreq(S) = nu:(s\ M) [ MC 5, [M] < a) (See Doignon, Reay, Sierksma [7] and Jamison [13].) - core1(S) D ---. for S C Rd.

Note that C(S) = coreo(S)

Kelly's theorem says that corel(s) 3‘ O provided [S[ > d + 2

The o—HeZZy number Me) of an aligned space (X,C) is the infimum

of all nomegative integers k such that

(b')

For each s in x with |s| > k + 1, corea(S) 3‘ 45

Thus h(o) = °° if there are arbitrarily large subsets of X with empty o-core. Note that h(l) = h, and that h(o) = 0 iff X = ‘9 or a = 0.

PROBLEM 1.

Find a formulation of the o-Helly number equivalent to (b') in

terms of intersections of collections of convex sets, similar to (a). The following theorem generalizes Levi's Theorem; a proof can be found in

[7].

THEOREM 2.1.

In any aligned space (X,C) the following holds:

h(o) 1 it follows that

m1n{|x|,h + o — 1} < h(o) < oh In the above theorem h is the l-Helly number.

It can be shown that both

bounds for 11(0) are sharp; see [7].

'

One of the most famous open problems in axiomatic convexity is the "sharpness" of the Eckhoff and Jamison inequality, namely r < c(h - 1) + 2, where I: is the Carathéodory number. Sierksma [23]. - c+2;

see

PROBLEM 2.

For a proof of this inequality see

For the 1—hdon number one can show that 1(1) < (1: - 1)ch

[7].

Find a relationship between r(1), h(1 — 1) and c, similar to the

Eckhoff and Jamison inequality.

Conjecture:

r('r) < ch(r - l) — c + 2.

The sharpness of the Eckhoff and Jamison theorem has stimulated the investi— gation of the product and sum spaces; see e.g., [8, 22, and 25].

Let (X1.Cl)

and (KTCZ) be aligned spaces; their aligned product space is the pair

(x1 x x2, 01902) where 01902: {A x a | AEcl, BE c2}.

GENERAL-IZATIONS 0F HELLY'S THEOREM THEOREM 2.5.

177

Let 011,01) and (X2,02) be nonempty aligned spaces with a-Helly

numbers h1(o) and h2(o), respectively, and assume that hi(o) + l < lxi| for i - 1, 2.

Then the 6-He11y number h(u) of (XI X X2, 01 63 02) satisfies

max(h1(a),h2(a)) < h(o) < a max{h1(u),h2(a)} — o2 + a

Note that for o - l we have h = max{h1,h2}, which is Theorem 8.3 of [23].

A

proof of the above theorem can be found in [7].

The concept of aligned sum space is introduced in Sierksma [25].

Let

(xl,c1) and 62,02) be aligned spaces; their aligned awn space is the pair (X1 UXZ, Cl + 02) where Cl + CZ = {(A\

BE 02}.

X2) U (B\

X1) U (An B)

I AE Cl,

when Xl= X2 we have 01+L‘2 ' {An B I A501, 3662} which is

the join alignment on X1 (= X2).

If X1 1'1 X2 = ¢ we have 6'1 + C2 ‘ {A U B I

A E 01, B E 02}.

THEOREM 2.6.

Let (X1,Cl) and (X2,02) be aligned spaces with X1 n x2 = d and

let h1(o) and h2 (a) be the respective o-Helly numbers.

Then the a-Helly

number 11(0) of (x1 U X2, 01 + (:2) satisfies

h(a) - 111(0) + h2(o) With the help of the aligned product and sum spaces it can be shown that

in case h < e - 1 the inequality

c < max(h,e — l}

is sharp where e is the exchange number; (see [22 and 25]). in case h > e — 1 is still open.

The sharpness

If e < c it can be shown that (see Sierksma

[25]): r 1+ the sharpness is still questionable.

Clearly,

the notions of aligned product and aligned sum space can be extended to prod— ucts and sums of more than two spaces with similar results for the o-Helly

. numbers.

178

SIERKSMA

PROBLEM 3.

Given the o-Helly numbers for the aligned spaces (X1,L‘1) and

(X ,Cz), what is the a-Helly number for the aligned sum space CZ), without the restriction X1 0 X2 = 4i?

(x1 U X2, (11 +

This problem is also open for the

Carathéodoty, Radon, and Exchange numbers.

Some partial results can be found

in Degreef [A]. PROBLEM A.

Let hl(a) and h2 (o) be the o—Helly numbers of (X,Cl) and 01,02),

respectively.

If 01 C 02, what is the relationship between h1(o) and h2 (a)?

This problem is also open for the other numbers.

In Kay and Womble

[16] the concept of C—halfspaces is introduced:

set H in X is called a C—haZf‘apace iff H E C and X\ H E C.

a

The aligned

spsce satisfies the separation property iff for each two disjoint convex

sets A and B there exists a C-halfspace H such that A C H and E C X\ H. Kay and Womble then proved the following characterization of h:

In any T1

aligned space having the separation property, the following two conditions are equivalent:

(X,C) has Helly number h < k;

(1) (11)

If s c x with |s| = k + 1 > 3, then there exists an element 1) e x such that every C—hslfspace containing at least 14: elements of S also contains p.

PROBLEM 5.

Find a similar charscteriztion of Mo).

Conjecture:

(ii)' h(o)

g k iff for s c x with [s] = k + 1 s 3 there exists M c x with- [MI = a such that every C—halfspace containing at least k elements of S also contains M.*

We conclude this section by giving the o—Helly numbers of several aligned spaces (with IX‘ = B“):

1. (x,(:) = (Rd,conv): h(a) = (d + 1):: 2.

C = (19.x):

3.

c= {A | Acx):

h(o) = o

a.

c= nouns | ACX, IA]2:

h(o) -s h(o) -k+o

*Recently J. P. Doignon has shown that the conjecture is false and that the correct characterization is:

(11)'

For 5 C x with |s| > k + 1 there exists an element 1: e x such that

every C—halfspace containing at least 1!. + 1 — 6 points of S also contains 1).

GENERALIZATIONS 0F HELLY‘S THEOREM

179

5.

X = Rd, C all closed sets in Rd:

Mo) =

6.

X . Rd, 0 all closed convex sets in Rd:

7.

Hf¢,0={AIMCA}:

a.

|u| = m> 2, c = {x} u {A | M s: A}: h(a) = max{m,0}

h(o) = (d + Do

h(o)=

An interesting aligned space occurs by taking the restriction of the ordinary alignment to the Gaussian integers (points in Rd with integer coefficients), introduced by J. 1’. Doignon [5]. see Jamison-Waldner [13]

PROBLEM 6.

He shows that h = 2“.

Also

for a different proof.

What is the a—Helly number of the aligned space (Rd,l,‘) where C

is the restriction of the ordinary alignment to the Gaussian integers of Rd.

Is h(a) = Zdo?* I 3.

ON THE DIMENSION OF THE a-CORE In [20, 21] .I. k. Reay has introduced the concept of (1,k)-partition:

Any set (family of elements) in Rd has a (mm-partition S - S1 U ... U ST if the sets Si are pairwise disjoint and dim[|'1:_1 conv(Si)] > k.

of (i=1 conv(Si) is called a (r,k)—divisible point of 5.

An element

Also see J. P.

Doignon [6].

FIGURE 1

By D (S) Reay denotes the set of elements p in x such that p 6 [111-1106 ) for some Radon T-partitions S= S1 U -- - U ST of 5. D D2(S) 3 ~ ~- .

Note that dC(S)=1D: (S)

According to Reay, the dimension of D (S) in Rd is the maxi—

mum of the dimensions of the intersections of the several Radon ‘r-partitions

of S.

For the five points in Fig. 1, din DZ(S) = 1 (S = {51’92’33’54’35D'

*Recently J. P. Doignon has shown that the conjecture is true, by tak— ing the 2(1 vertices of a fundamental cube in Z and 0‘ 1 more lattice points on the 2d "symnetric rays" starting in the vertices of the cube; these 52d points have nonempty c—core.

SIERKSMA Note that dim coral-(S)

meow 3.1.

-

180 2, and that 112(5) S. core1(S).

For any aligned space (x,z:) and s in x the following holds:

C(D1(S)) C core1_1(S)

As we have seen above, the converse of this theorem is not true in general.

The proof of 3.1 is a direct consequence of 2.1.

similar to Birch's, Reay

THEOREM 3.2.

Using an argument

[21] has shown the following theorem.

If (X,C) - (Rz,conv) and S C R2 then

conv DT(S) = core1_1(5)

As is customary,

the dimension of a subset in Rd is taken as the dimen-

sion of its affine hull.

PROBLEM 7.

Three additional problems on the c-core follow.

Find s convexity space (X,C) such that for some n0 > 0 and each

n > no there is a set s in x with [S] - n and 0031(5)) ,4 core1_1(s).

PROBLEM a. (d + 1) + 17

Is conv DT(S) - core1_1(S) for each s in Rd with |s| - (1 — 1) Note that if this is the case, we have an alternative proof of

Twerberg ' s Theorem.

PROBLEM 9.

Is the number of Radon 1-pertitions dependent on the number of

vertices of core1_1(s) for finite sets S in Rd?

I.e., for the o-core being

a single point does the set have the minimum number of Radon 1-partitione'!

THEOREM 3.3.

For each s in Rd with 13] > Zad + 2,

din: corea(S) > 1 The proof follows directly from Reay [21] Theorem 5 together with 3.2. This theorem also says that each set in R“1 with at least 20d + 2 elements admits at least two elements in the a-core.

That 20d + 2 is the best pos-

sible result follows from the fact that Zod + 1 points with a points on each of the positive as well as on the negative axes together with the origin have as o-core the origin, so its o—core is 0-dimensional.

GENERALiZATIONS 0F HELLV'S THEOREM

131

In the‘following theorem we answer the question: number of points with a d-dimensional o—core.

What is the minimum

H+ and 11— are the open half—-

spaces determined by the hyperplane H.

THEOREM 3.4.

(i)

Let S C Rd.

Then (i) ’ (ii) ° (iii), with:

din corea(S) = d;

(11)

I

For each hyperplane H in Rd, IH+ 0 SI > o + 1, or |H_ n SI > a + 1, or both;

(iii)

ls! > 2n + d + 1.

Proof.

Take any 5 in Rd.

(1) = (11): Take any hyperplane H in Rd with |H+ n s] < o and |H' n s| < d.

Then, clearly, corea(S) C H, and therefore, dim corea(S) < dim H = d - 1,

. which is a contradiction.

Hence, each hyperplane has at least a + 1 points

of S on at least one (open) side. (ii) 9 (iii):

If S has at most Zo + d elements, there is a hyperplane

H spanned by at least d elements of S and with at most I! elements in each

open halfspece 11+ and 11’. This contradicts (ii). Therefore |s| > 2o + d + 1. D An element 1) in X is called an extreme element of S iff p e conv(S \ p). Also see the next section for s more extensive use of extreme elements and sets in relation to the a-core.

THEOREM 3.5. points.

Let s C R2 with ls] > 2s + 3 and with at least 3o + 1 extreme

Then dim coreu(S) = 2.

The proof of this theor— is omitted because it is similar to the proof of the next theorem. The example in Fig.

2 shows that for o x 2 we need in fact more than 6

extreme points in the above theorem; note that the Z-core is the point "7." The independence condition in Theorem 3.6(11) plays a part in the next theorem.

In the proof is used Peterson's method of projections; see Peter—

son [19] and also Doignon [6].

Note that the smallest sets for which (ii)

holds with o = 0 are aimpliaes.

'1‘!!m 3.6. Let s c R3 with 151 > loo + 3 and with at least he + 1 extreme points; moreover, suppose that for each hyperplane H, or |H+ '1 S| > o + 1,

.0! [li— n SI > o + 1, or both are.

Then, dim corea(5) = 3.

51 ERKSMA

182 FIGURE 2

Proof.

Take any sets T1, T2, T3. TA in s with l'ril < o for 1 = 1.2.3.1..

A Then IU1=1 Ti l

_

< 4o, so there is an extreme point p in S with p e

Let 11 be a hyperplane with p G 1-1

+

and conv(S \ p) C H .

1-1 0 ([p,s] I s E S \ 1)}, hence, IS*I > 1m + 2. dim corea(S*) > 1.

1-1 Ti'

Then, define 5* =

Theorem 3.3 then implies that

Clearly, the dimension of the o-core of 5* cannot be e-

qual to 1, because, otherwise, there would exist a line L with coreaS“ C L

in 1-1 with less than o + 1 points on both sides, and then the hyperplane aff(p U L) has less than u + l on both sides, which contradicts the assumption of the theorem.

Therefore, dim core°(S*) = 2, and so we have (17} U

corec(s*) C “2-1 conv(S \ Ti)'

Grunbaum [11] then implies that dim corea(S)

= 3. D

The above theorem makes use of the first part of Grunbaum's theorem in

the paper "The Dimension of Intersections of Convex Sets" (1962).

Katohalski

(1971) has pointed out in his paper with the same title that the second part of Grunhaum's theorem is wrong; see [lb and 15]. Ketchnlski's generalization of Kelly's theorem.

The following theorem is The number o(k,d) is defined

as follows:

u(k,d)-d+l

ifk=0

u(k,d) - msx{d + 1,2d - 2k + 2) THEOREM 3.7.

if 1 < k < d

If G is any finite family of convex subsets of Rd, and for

every subfamily F with |F| < u(k,d), dim n? > k, then am no > k.

GENEMLIZATIONS 0F HELLV'S THEOREM

183

Let c k + 1 is called k—independent (see e.g., [21]) iii for each 1‘ in s with [TI = k + 1, dim aff(T) = k.

Note

that each nonempty set is 0-1ndependent and that "cl—independent" 1s equivalent to "in general position. "

mom 3.3. Lero k

Proof. Take any 1: (= o(k,d)) sets T1.,,,.Ta in s with ITiI < o for each 1 - 1,...,n. As Isl > ou(k,d) + k + 1, and lug:1 Ti] < ou(k,d), it follows

that Is \ ”i=1 Til > k + 1. As 5 is k-independent, it follows that dim aff(S \ kfil=1 Ti) > k. ly, S \ U;=1 Ti C ”i=1 (5 \ T1) C “1:1 conv(S \ Ti)" (S \ T1) > dim(S \ d;=1 T1) > k.

Clear-

Hence, dim 02:1 conv

Katchalskl's theorem then implies that

dim coteU(S) = dim n(conv(S) \ T) | T c s, |'r[ < o) > k. a Note that for k = 0 in the above theorem the condition "S is k—independent" can be deleted; the theorem then reduces to Theorem 2.2.

orem also holds without "ls—independent"; see Theorem 3.3.

For k = l the the-

In the following

corollary k = d; also see Reay [21], Theorem A.

COROLLARY 3.9. tion.

Let S C Rd with [S] > (o + 1) (d + 1) and S in general posi-

Then

dim core°(S) = d

The conditions on the set S in Theorem 3.8 are not necessary. for instance, the seven points in the plane as shown in Pig. 3.

Consider

It is clear

that the dimension of the 2-core of these seven points is 2, although there are just seven points.

Note that the points are in general position.

For the case n - 1 several more theorems can be obtained by taking into account Theorem 3.1 and making use of the theorems on (2,k)-divlsib111ty in

Doignon [6] and Reay IN, 21].

On the other hand, it is remarkable that the

theorems on the o-Helly number and on the dimension of the o—core are generally not very difficult to prove, whereas the similar assertions on the t-Radon

SIERKSMA

184 FIGURE 3

(Eckhaff'a conjecture) and on the dimension of (TAO—divisible points are to a great extent still open, although there is a quite strong relationship

between the Kelly and Radon relatives.

Is the main reason for this gap in

difficulty the fact that the o-core is convex and the set of (1,k)-diviaible points are not convex in general?

PROBLEM 10.

Find necessary and sufficient conditions for the o-core of a

set S in Rd to be k—dimensionsl (0 < k < d).

4.

REAY— AND LARMAN-TVPE GENERALIZATIONS 0F HELLY'S THEOREM . In Reay [21] a generalization of Radon's and Tverberg's theorem is given

in the ordinary aligned space (Rd, conv).

Along this line we give the fol-

lowing generalization of the Radon number. Suppose 11,1 > 1 are integers.

The (r,n)-R:zdon nwrbar r(1,n) of an a-

ligned space (X,C) is for n < r the infimum of all positive integers k such

that for each s c x with Isl > k there is a 1-part1tion s - s u 1

U 51

of S into pairwise disjoint sets with the property that

C(51)” "' n C(Si ) f¢ l

n

for each 11,...,1n E (1,...,'t}; for n > 'r we define r(1,n) I r(1,1).

Note

that r(1,1) is the T-Radon number (Tverberg's generalization), and that

GENERALIZATIONS 0F HELLY'S THEOREM

185

r(1,1) = T.

For a generalization of Levi's theorem 2.1 similar to Reay's generali— zation of the Radon and 'i‘verberg theorem, we introduce for 11,0 > 0 the (0,11)—

Helly number h(a,n) of an aligned space (X,C) as the infimum of all nonnega—

tive integers k such that each s c x with Isl ; k + 1 has the property that C(S\ M1) n ---. nC(S\ Mn) #¢

for each "1'”"Mn in S with 1141] < a and each i = 1,...,n. and [z

= a', then h(o,o') is the a-Helly number.

If ISI =p > a

Furthermore, h(o,l) = a.

Levi's theorem 2.1 can be generalized as follows. THEOREM lo.1.

For any aligned space (X,C) the following holds:

1.

Mom) < min{na,h(o)}

2.

h(o,n) = h(a) < oh(l)

3.

h(o,n) < no < uh(1)

if n > h(l) if n < 11(1)

Proof. (1) Take any 8 c x with N > 1 + min{no,h(a)}. If |s| > 1 + 110:), theno I corec(S) C “{C(S \ Rik) | nik c s, Inik| < a for k = 1,...,n} and hence h(o,n) < h(o).

If [S] > 1 + no then, clearly, for each n sets

M1""'Mn in s with 1141] < a there is an element 1) E s such that p 6 s \ (M1 u --- ”“19 = (s\ M1) n n (5\ Mn), so that n{C(s\ Mi) [1 = 1,...,n} # ¢ and it follows that h(a,n) < nu.

Hence, h(a,n) < min{na,h(a)).

(2) Can—

side: the family F - {c(s \ n) | M c s, IMI < a}. As n > h(1) it follows that for each F0 c F with [F0] = h(1), "F0 3‘ ¢. Helly's theorem then implies 0F f 0.

Hence, h(a) < h(o,n).

yields the desired result.

(3)

Combining this with (1) and Theorem 2.4 For n < 11(1) it follows from (1) that

h(o,n) < [10 < uh(l). I: The following theorem is a generalization of Levi's theorem for the

(a,n)—Helly and (hm-Radon numbers.

THEOREM A.Z

In any aligned space (X,C) the following holds:

h(a,n) m.

If a finite space x has no Radon

1-partition (so that :(1) = |x| + 1) then the theorem again is trivial, since h(o,n) < lxl for any a > 0. Thus we may assume a > 1 and r(a+l, n) is finite First let n < a + 1.

and is at most IXI in case X is a finite set.

Take any

5 c x with IS] = r(o+l, n) and let ml....,nn be n subsets of s with |n1| < o for i - 1,...,n. Furthermore, let S - Sl U '-- U S 0+1 be a (a + l)-patt1tion of S with the property that for each n of its sets the convex hulls have nonempty intersection, i.e., C(S

2,...,a + 1}.

11

) n 0' n C(S

For each Hi there is a set SJ.

that Sji C S \ “1 .

Hence, C(Sjl)

in

) 1‘ fi for each i ,...,i 1

n

E (l,

with ;l1 6 {l,...,a + 1} and such

1 C C(S \ H1), ... , C(Sj ) C C(S\ Mn), and

n therefore we have “i=1 C(Sj ) C “i=1 C(S \ Hi).

As nil-1 C(S

l

jl

) 1‘ O, it fol-

lows that n‘i‘_1 (:(s \ Hi) ,4 ¢. This implies that in fact h(a,n) < rum, n) - 1.

For n > a + 1, it follows from 11.1 that h(o,n) < h(a) < r(a+1) - l I

r(o+1, 0+1) — 1 = r(|1+1, n) — 1. u THEOREM 4.3.

For the ordinary convexity space with dimension d the following

holds: h(a,n) = min(nu, (d + Do) Proof.

Ifn> d + l = h (1), then 5.1(2) implies h(a,n) =. (d + Do.

Now let n < d + 1.

Take some simplex in R'1 and let S be the set consisting

of n vertices of a simplex with multiplicity a; let H1,...,Hn be the vertices

with multiplicities a (S is a family of elements).

Then, clearly, conv(S\ H1)

0 '-- n conv(S \ Mn) = ¢, and therefore h(a,n) > no. follows from (1) that h(o,n) = no.

Let S C X with |s| - m > o. of the

m

fewer than

The o—core of S is then the intersection

sets C(S \ M) with H C S and [HI = a. I;

sets.

As no < (d + Do it

Hence, h(a,n) = min{na,(d + Do). U

In general one can take

By mr we shall mean the minimum number of sets Mi in

s with Inil = G such that corea(5) = C(s \ M1) n

n C(S \ Mn*).

He now pay some attention to the nature of the sets “1 that are needed to construct the o-core.

To that end we introduce the following concepts.

Let S be a subset of X.

A subset T of S is called an extreme set of S if

GENERALIZATIONS 0F HELLY'S THEOREM

187

1‘ 1" C(S \ T) ' ¢

An element p of S is called an extreme element (point) of 5 iff {p} is an extreme set of S. The set '1‘ is called an extreme cluster of S iff

C(T)nc(s\ '1’) -¢ In the following figure 5 - {s1,...,s6] U S7 with 157] = 5. is an extreme set as well as an extreme cluster in S.

extreme but not an extreme cluster.

FIGURE b

The set 57

The set (57’36) is

The element 53 is not extreme.

5'

S

0n the other hand consider the aligned space (X,C) with C - (t) U {A I

A C x, a E A} for some fixed element a E X. and for each a > 1 we have coreo(s) - {a}.

Then for each 5 C X with IS] > a If a E S then each subset of S

is extreme; if a E S then the Only subset of S that is not extreme is (a). Note that there are no extreme clusters. Note that an "extreme cluster" is always "extreme."

Also note that both

components of a Radon 2-partitinn of a set are extreme clusters, i.e., T is

an extreme cluster of 5 iff S \ T is an extreme cluster of 5.

By 210(5) we

denote the collection of all extreme sets in S with at most 0 elements.

THEOREM 5.4.

Let (LC) be an aligned space and S C x with [S] > :1.

Then

the following holds: corea(s) - ”(C(S \ E) I E E 35(5)}

Proof.

Take any s In x with |s| > a, and let M be a subset of s with

133

SIERKSMA

IM| < o. Define M = M1 u E1 with M1 c C(S \ n) and E1 n C(S \ H) = ¢. Clear1y, E1 6 55(5), and c(s\ n) = c(s\ (141 U 31)) = c(s\ M1)\ E1) c C(S \ E1). Hence, corea(S) - n{c(s\ M) 1 MC 5, IM] < a} c n{c(s\ E) | E s 50(5)}, As the converse of this inclusion holds trivially, we have that in fact

corea(S) =n{c(s \ E) | E 6 120(3)}. El The U—cure of infinite sets in Rd is somewhat different from the o—core of a finite set in Rd.

Consider for instance the l—core of the closed can-

vex circle disk C and the closed square D in R2. equal to the a-core of C or D for each a > 1.

The l-core of C or D is

Note that the o-core of C is

the open circle disk and that the a-core of D is D minus the four vertices. In any case, the a—core is the set itself minus its extreme points.

The fol-

lowing theorem characterizes the o—core of those "infinite" sets. FIGURE 5

THEOREM 5.5.

Let (x,c') be an aligned space and s c x with |s| > 5.

Then

the following assertions are equivalent: (i) (ii) Proof. E)

06‘) \ E = C(S \ E) for each E 6 190(5); coreu(S) - 0(5) \ UEU(S). (1) = (ii):

C(S) \ UEU(S) = “(0(5) \ E [ E E 175(5)) - “(C(S \

I E E 120(5)) = corea(S) (according to LA).

(ii) 9 (1):

As C(S \ E) U E C C(S) for each E C S, it suffices to shew

that C(S) \ E C C(S\ E). U [UE°(S)].

Take any p 5 0(8) \ E.

Then, 1) E C(S) - [cotea(5)]

If p E UE°(S) C S, then 1) E S\ E C C(S \ E), and we are done.

If p E core6(s), it follows from 4.4 that p E C(S'\ E).

In either case

C(S) \ EC C(S\ E). El Let S C X with |s| > o,

By (115(5) we denote the set of all cluster ex-

GENERALIZATIONS 0F HELLV'S THEOREM

189

treme sets in S with o elements, i.e., E 6 015(5) iff C(E) n (I(S\ E) = o

and Isl - a.

Note that cza(s) c saw).

THEOREM 4.5. Let (x,c) = (Rd,conv) and s c Rd with ;s| = m > :1. Then the following holds 1.

coreU(S) - n{conv(5\ E)

2.

n*(a,s) = ICZU(S)I is the number of non-Radon 2-psrtitions of s

I E E 010(8)}

with one component consisting of 0 elements

Proof.

(1)

According to 4.5 we may restrict the proof to extreme sets.

Let M C S be an extreme set with IMI = a.

M can be partitioned into pair-

wise disjoint sets, say M = M1 U --- u Hk’ such that each “i is contained in

an extreme cluster of S.

As conv(S\ H) = conv(s \ (U1;=1 141)) = conv(n:=l(s

.\ 141)) - ”11:1 conv(S \ Mi) it follows that n{conv(s \ E) | s 6 {710(5)} C n{conv(S \ n) | ME 311(5)} = corea(S). hence, (1) holds. (2) is a direct consequence of (1) and the definition of extreme cluster set. D

For the number n*(a,s) - |czq(s)| with |s| - m > a > 1 in 4.6 it fol— lows that:

n*(o,s) < [5'] - r

a

where ra is the a-component of the Radon vector

(:1, . . . Jinn/2])

of s; if ls] = 1, then ri is the number of Radon 2-partitions of 5, say {51,52}, such that [Sll = a and ISZI = i - a. For a survey of results on It is still an open problem to give an in-

Radon vectors, see Eckhuff [9].

trinsic characterization of the Radon vector (r1....,r[m/2]).

PROBLEM 11.

Find an intrinsic characterization of the number n*(a,S) , where

ISI = n > a > 1.

Is n*(o,s) -

I:

- ra if S is in general position?

that the first part .of this problem is equivalent to [9] Problem 3.1.

THEOREM 4.7.

For any aligned space (X,C) the following holds:

Note

190

SIERKSMA

max(n | 11(0) = h(a.n)} k

for each M1,...,Mn in s with I341] < a and each i e {1,...,n}.

Note that

H(d,o,n,0) is the (o,n)—Helly number of (Rd,conv). The following theorem is another generalization of Levi's theorem; the proof is similar to the previous generalizations; see Theorem 4.2. TREORH‘I 4.9. PROBLEM 12.

H(d,a,n,k) < T(d,U + 1,n,k) - l. Calculate H(d,a,n,k) under certain independence conditions, e.g.,

k-independence; see e.g., [6].

Does equality hold in 4.6?

In Larman [17] Radon partitions are studied for sets in Rd where some of its points are "stolen."

Along this line the Laman—Radon and Lsrmsn—

Belly numbers are introduced below.

Let v be an integer > 0.

I

*

The (1,v)—Iaman-Radan number LR(-r,v) of an

GENERALIZATIONS 0F HELLV'S THEOREM

191

aligned space (X,(,') is the infimum of all positive integers k for which it

is true that for each set S in X with ISI > k and each set '1' in S with |T| < v:

there exists a T-partition S = $1 U ... U ST such that

C(sl\1‘)n-~.nc(s \ T) #6‘ For (Rd,conv) the number LR(d,I,n,k,v) may be defined similar to the number T(d,'r,n,k). The (o,v)-Lamzm-HsZZy number LH(a,v) of an aligned space (X,C) is de—

fined as the infimum of all nonnegative integers k such that each S in X with

IS] > k + l and each T in S with IT] < v have the property that

l'1{L'(s\ (u\ T)) lmcs, IMI 0 included in S and let d be the semidistance associated with p.

Assume for simplicity of notation that the center of E is 0.

The radius r will be chosen small enough in such a way that

{y 6 V2 2 F 0 such that ix 5 B,

¢(Ax) = A¢(x)

In fact, it suffices to prove this equality for any x E B and A 6 [0,1]. proof is a modification of that of Lemma 1, p. 142 of [2]. Let p' E P

1

and let d' be the semidistance associated with 11'.

Set

E1 - (y e s : d'(0,y) = d'(y,X) = % mm) and mm < r} and, if En has been defined, let D'(En) he the d'-diameter of En and set

_ {y 5 En .. d v (y,x) < En+1 — '

was“) 2

,

Vz E En}

It is easy to see that if EM1 !‘ 4', D I (En+l)
S 2“ I E henceZEEn+11f ZEEn. Now since

tux-l) = P(x - %) = 1:0: - %) = 12%) = % p(X) < t and

me ,2i = d'é2, x) = iZ d'(0 1 x) § 6 E1, and therefore, by induction,

15 n a 2 n=1 En “17'

3—.

“Agree 4

the d'-diameter of “17' is obviausly zero.

UNIMORPHIES 0F SUBSETS

197

x E Since _ 2 MP, for every p I E P1,

x E — 2

where M has d'-dismeter zero for every p' E P1, hence is reduced to a point: x

u = {i}'

The image of B under ¢ is the semiball

{x E v2 : F(p) (x) < r} and it is readily seen that, for every pi E P1

M‘Np') = 604p.) - where "17(11') has been constructed from ¢(x) like above.

Therefore,

it 12 = 4—H Zx In fact, we have shown that, if X1

is the center of a sufficiently small

closed p-semiball included in S and if x2 is a point of this semiball, the image under 4' of the midpoint of [xl:x2] is the midpoint of [‘fi (x1):¢ (x2)]. New, starting with 0 and % for new x1 and x2 respectively, one gets

¢(%) = % ¢(x) and, since the semiball

{y = P(%-y) n.

this question is given by Eberhard's Theorem [1, lo]:

209

The answer to

210

MALKEVITCH

THEOREM 1 (V. Eberhard).

If pg, p2, pg, P?! ..., p: is s nonnegetive integer

solution of (1), then there exists a nonnegative integer p2 and a convex 3valent 3—polytope P such that pk(P) - p]: for k, 3 < k < n, and pk(P) - 0 for k>n.

For proofs see

[1, 4,

7].

All proofs of this statement require careful

"bookkeeping" in the construction of an appropriate plane, 3—connected 3valent graph, on which Steinitz'l Theorem [lo] can be invoked, thereby guaranteeing the existence of the appropriate appolytope.

Steinitz's Theorem states

that necessary and sufficient conditions that a graph G be isomorphic to the graph of vertices and edges of a convex 3—polytope is that G be planar and 3—connected.

The spirit of proof of an Eberhard type theorem is given by the

much more transparent proof of a generalization of Eherhard's Theorem to ob— tsin s 4-valent analogue, due to Branko Grunbaum.

THEOREM 2 (B. Grflnbaum).

If p5, pg, pg, ..., p; is a nonnegative integer

solution of equation (2):

p3 = 8 + Ek>5 (k — wk

(2)

then there exists a nonnegntive integer PZ and a convex 4-valent 3-p01ytope P such that pk(P) = [21: for 3 < k < n, and pk(P) = 0 for k > n.

The proof of this result is very elegant and appears in [A].

Ernst

Jucovic [7] gives another proof of this result and proves the following ex— tension. THEOREM 3 (E. Jucovio).

For every sequence (p5, pt, ..., 1);) satisfying (1)

and with p: > {3:5 (3k — 10),)? there exists a 3-polytope P with pkfl’) - pg, 3 < k < n.

T. C. Enne [2] has recently strengthened this theorem.

The purpose of

this note is to give a new proof of Theorem 2 and a result in the spirit of Theorem 3. The proof will exploit an extension of an idea developed in [9, 11] and used to solve a problem in [5].

EBERHARD'S THEOREM FOR 4-VALENT POLYTOPES

211

Let G he a plane k—vslent (k = 3, k, or 5) 2-connected graph in a fixed embedding in the plane. By a k-ahell diagram of length m is meant a plane

simple circuit of length m with (k - 2) short spikes into the interior and/or exterior of points on the m circuit.

Figure 1 shows two 3—she11 diagrams of

length 6, for k = 3. The exterior code of a 3-shell diagram is the number of outspikea at each point of the diagram.

The codes for the diagrams in Fig.

and (l,1,0,1,1,0), respectively.

l are (1,1,0,0,1,0)

Note we treat the code cyclically, and it

could be considered to start in any position. ' Two codes are the same if they are identical when read in either cyclic order.

The interior code of a shell

diagram is the number of inspikes at each point of the diagram.

The interior

codes for the diagram in Fig. l are (0,0,l,l,0,l) and (0,0,l,0,0,l), respec— tively.

If the interior code and exterior codes are identical, the 3—she11

diagram is called self‘invertihle or invertible.

The diagram on the left in

Fig. 1 is invertible but the one on the right is not.

The 3—she11 diagram

on the left in Fig. 2 also has the property that it is replicable by a single

hexagon.

This is shown in Fig. 2.

After a single hexagon is adjoined, the

resulting shell diagram is the same as the original one. The wiggled circuit is the new shell diagram, which has the same codes

as the original.

To get a new proof of Theorem 2, we will use the fact that

Fig. 3 shows a A—ahell diagram of even length (which is invertible) and which is replicable by a single A-gon.

Using the existence of this 4-shell, we see that if there exists a planar lo-valent, 3-connected graph G which contains such a shell, then there also

exist A-valent 3-connected graphs Hm with m A-gons for all values of m which exceed 1),. (G) .

FIGURE 1

212

MALKEVITCH

FIGURE 2

CH? To Shaw how to get a new proof of Theorem 2, we first show that the in—

terior of a lo-shell of even length can have its interior completed with faces which are only 3—gons and A—gons.

The construction is shown in Fig. 3.

The

diagram on the right shaws, with the dotted lines, how to increase the length of the shell by 2, and create only. new k—gons.

For a solution 0f (2) with faces of l-gth > 5, we proceed as follows. The general case, is done in the same fashion as the example.

To create an

S—gon, S—gon, and 6-gon see Fig. 4.

Note that for each k—gon, (k > 5) k - b triangles are created, as required by (2).

Furthermore, there are A additional triangles; needed to

"close up" with the extra 4 triangles from Fig. I. to get the eight triangles FIGURE 3

EBERHARD'S' THEOREM FOR 4-VALENT POLVTOPES

213

FIGURE 4

66» required in (2).

The shells in Fig. la can be closed with one such as in Fig.

5', and are replicable with one A—gon.

Since this construction yields lo—

valent plane, 3-connected graphs, Steinitz's Theorem can be involved to complete the proof.

The number p: obtained by this construction is not minimal for a given solution of (2).

The proper order of magnitude for a minimal value of p2

is found in T. C. Enns' paper [2].

It is interesting to note that our proof

is in the spirit of Theorem 3, in that all values of 1),. beyond the one in the construction are attainable.

This is not the case for (1).

For the solution

p5 = A of (1), only even pg allow a polytope to be constructed. If one writes down the Euler type relation for the valences ti of a tree , one obtains

FIGURE 5

214

MALKEVITCH c1=2+21>2 (i—Z):i

(3)

The fact that the coefficient of t

vanishes suggests that one might be able 2 to prove an analogue of Eberhard's Theorem for this case. This turns out to

be the case.

Details can be found in [10].

REFERENCES l.

V. Eberhard, Zur Morphologie d22- Polyeder, Teubner, Leipzig, 1891.

2. T. Enns, Convex 4-va1ent polytopes, Discrete Math. 30(1980), 229—234. 3.

J. C. Fisher, An existence theorem for simple convex polyhedrs, Discrete Math. 7(1974), 75-87.

A.

B.

5.

B. Grijnhamn, Some analogues of Eberhard‘a Theorem on convex polytopes, Israel J. Math. 6(1968), 395-511.

Grfinbaum, Convex Palytopes, Interscience, London, 1976.

6.

E. Griinbaum and J. Zeke, 0n the existence of certain planar maps, Dis— crate Math. 10(1974), 93-115.

7.

S. Jendrol, A new proof of Eherhard's Theorem, Acta Fae. limit). Comesmlence Math. 31(1975), 1-9.

B.

E. Jucovic, 0n the face vector of s A—valent, 3—polytope. Stud. Sci. Math. Hung. 5(1973). 53—57.

9.

J. Malkevitch, Properties of Planar Graphs with Uniform Vertex and Face Structure, Ph.D. Thesis, University of Wisconsin, Madison, Wisconsin,

1969. 10.

J. Malkevitch, Spanning trees in polytopal graphs, Arm. N. 3. Academy of Science 319(1979), 361-367.

ll.

.1. Malkevitch, 3—valent, 3-polytopes with faces having fewer than 7 edges, Arm. N. I. Acaderry of Science 175(1970), 285-286.

GRAPHICAL DIFFERENCE SETS AND PROJECTIVE PLANES* Andrew Sobczyk Department of Mathematical Sciences Clemson vmiversity Clemson, South Carolina

'I.

INTRODUCTION One justification for presentation of a paper in combinatorics at a con-

ference on convexity is of course the combinatorial structure of convex polytopes.

In space Rd of dimension d = 6k or d = 6k + 2,

the skeleton—edge struc-

ture of a simplex is a Steiner system on 6k + l or 6k + 3 vertices; i.e., there are sets of edge—disjoint 2—faces or triangles which cover all of the

edges, each exactly once.

At the problem sessions of this conference, the

author will pose questions like the following:

Associate K15, the complete

graph on fifteen vertices, with a regular planar polygon (see Fig. 1).

The

edges are classified, by numbers of vertices "skipped," as 0, l, 2, 3, 4, 5, 6 E 7.

All skip-Z edges are covered by the three equiskip—Z S-circuits,

and the remaining skips by two rotational series of fifteen triangles each,

the skip 0, 3, 4 triangles and the skip 1, 5, 6 triangles.

Is there a 13-

polytope having fifteen vertices, which correspondingly has thirty triangular

2-faees and three pentagonal 2-fsces, all edge-disjoint?

(The edges of K15

are uniquely covered also by the three 5-circuits and by fifteen quadruples with outer skips 0, 3, l, 7 E 6, inner skips A, 5.

*Presented by title

215

In other terms, the

216

SOECZYK

FIGURE 1

nonskip—Z edges are covered by the fifteen skip 0, 3, 4 triangles, and the fifteen 1, 5, 6 "forks" or basic matroid eluents of K15.) Here is another question of a similar nature:

The edges of K

are

uniquely covered by two series of thirty-nine quadruples, those wig?! respective exterior skips 0, 17,

8, 10; 1, 13, 16, 5, by thirteen equiskip—lZ tri—

angles, by the thirty-nine skip 2, 3, 6 triangles, and by the thirty—nine skip 4, 9, 14 triangles.

Is there a polytope, in Rd of appropriate maximal

possible dimension d, which correspondingly has seventy-eight rectangular 2— faces and ninety—one triangular 2—faces?

Perhaps not all of the triples and

quadruples can correspond to outer 2-faces; if so, this would reduce the

maximum possible dimension d.

Skips for the several series may be clockwise

(c.) or counterclockwise (cc.);

is it possible that there are polytopes for

some combinations (c., cc., cc., c., etc.) but not for others? Also,

the author believes that a viable topic is convex sets in the

vector spaces over the Galois fields GF(p) and GF(pk) [p prime, k an integer > 1].

The points of the affine plane AG(2,pk) are the ordered pairs (my),

x, y E GF(pk) [k > 1].

As usual, the lines have equations of the form y =

mx + b, or x = c in case of infinite slope, m, b, c e GF(pk). Thus if n — pk, AG(2,n) has n2 points, and the n2 + 11 lines fall ,into (n + l) bundles of n parallel lines each.

For each point P(x,y), there is a pencil of (n + 1)

lines shich pass through P.

As soon as the author establishes the existence

GRAPHICAL DIFFERENCE SETS AND PROJECTIVE PLANES

217

of finitekvector spaces Rd which are not the sets of d-tuplea

(X1, ...,Xd ),

Xi E GF(Pk) but which otherwise have the usual geometric properties of affine Rd (1. e., of finite affine spaces of orders different than pk), the "convex-

setters" (axiomatic and nonaxiomatic) may go to work in such spaces! In view of results quoted, e.g., on pp. 27—28 of [2], with the usual axioms for affine and projective spaces,

for d $ 2 there are no affine or pro-

jective d—spaces other than Galois (= Desargueaian).

An (unusual) affine

d—spaoe, d f 2, such as the author Visualizes, might involve matroids [6], nearfields, ternary rings, neofields [5], or‘other generalizations of Galois fields.

To extend AG(2,n) to be the projective plane PG(Z,n), as usual for each slope there is adjoined a point at infinity; the line at infinity consists of all the points at infinity.

Thus each line of PG(2,n) has (n + 1) points,

and the total number of points, as well as the total number of lines, becomes

_(n2 + n + 1).

For n other than n = pk, the existence of AG(2,n) is equiva—

lent to the existence of a corresponding PG(2,n).

2.

ROTATIONAL MODELS FOR PROJECTIVE PLANES A line contains all of its segments; in that respect a line having (n +1)

points is analogous to an n—simplex.

The projective plane of order 2 may be

viewed as the rotational Steiner system on seven vertices.

The edges of K7

are uniquely covered by the seven skip 0, l, 2 triangles, which are the seven

lines of PG(2,2).

The six lines of AG(2,2) may be regarded as the six edges

of KA' A rotational model for PG(2,3) is provided by K13 associated with a

polygon having thirteen vertices, the lines being the thirteen quadruples which have outer skips l, 0, 3, 5,

inner skips 2, A.

The quadruples uniquely

cover all edges of K13. There are similar rotational models for all the projective planes PG(2,n)

over GF(pk) [n - pk], at least for n = 4, 5. 7, a, 9, 11. For 11 - A, the

twenty—one lines are the twenty-one quintuples of lg,1 having outer skips 2, For n = 5, the thirty—one lines (111

0, 4, l, 9, inner skips 5, 8, 6, 7, 3_.

K31) are the thirty-one sextuplea having outer skips 3, 5, 12, 0, 1, 4, inner skips 8, 9, 18 E 11, 13, 2, 6, 7, 10, 15 E 14. For n = 7, the outer skips are

4, 0, 1, 9, 18, 3, 6, 8, and inner skips the remaining skips of the skips 0 >through 27 E 28 of K57; for n = 8, the outer skips in K73 are 0, l, 3, 7, 15,

218

SOBCZVK

A, 17, 8, 9; for n = 9, in K91 they are 0,1, 5, 17, 21, 6, 4, 1s, 3, 9; for n = 11, in K133 they are 6, o, 1, 13, 11, 31, 1s, 5, 1., 3, 17, 12. 3.

ROTATIONAL MODELS FOR AFFINE PLANES The Steiner system on K9, which is AG(2,3), may be presented as the

eight skip O, l, 2 triangles of K , together with the four triangles formed by joining an exterior vertex a (Say a is the origin of AG(2,3)) with the four skip—3 (diagonal) edges of K8.

If the vertices of K8 are numbered cy-

clically as 1 through 8, the eight skip 0, 1, 2 triangles are 124, 235, 346, 457, 568, 671, 782, 813. AG(2,3):

e.g., lines 124,

It is easy to verify that the Steiner system is 568, “37 are a typical bundle of three parallel

lines; the triangles which include a as a vertex are the pencil of four lines through a; and through each point of K

there is a pencil of four lines, e.g.,

for vertex 1, the lines 121», 671, 813,861.5Following is a different model for AG(2,3).

Present the Steiner system

on [(9 as the join of an exterior triangle (line of AG(2,3)) and K6'

The equi—

skip-l triangles of K6 are the two parallel lines to the exterior line.

The

three skip—Z diagonals of [(6, together with two disjoint sets of three of the skip-0 edges, are three sets of three disjoint edges; the remaining nine lines are the joins of each of the vertices of the exterior triangle to the edges of one of the disjoint sets. The affine planes AG(2,h)

and AG(2,5) have models with exterior point

a, like the first given above for AG(2,3).

For AG(2,A), five of the twenty

lines are the joins of exterior vertex o to the five equiskip-4 triangles of K15.

The remaining fifteen lines are the fifteen quadruples which have ex—

terior skips 3, "l, 0, 7 E 6, inner skips 2, 5.

For AG(2,5), o joined to the

six equiskip—S quadruples of K2,. forms a pencil of six lines through a.

The

remaining twenty-four lines are the twenty—four quintuples of K24 with exterior skips 0, 2, 4, 1, 10, inner skips 3, 8, 9, 7, 6.

(Skips 5, 11 are

covered by the equiskip-S quadruples.) The affine plane AG(2,5) , like AG(2,3) , also has a model formed by join— ing an exterior line of 5 points to K20.

The four parallel lines to the ex—

terior line are the four equiskip—3 quintuples of K

11.

, they cover edges 7,

Five more lines are the five equiskip-lo quadrufiges with their points of

intersection with the exterior line.

The remaining-twenty lines are the

twenty quadruples of K20 which have outer skips (19 E B), l, 0, 5, inner skips 2, 6, joined to their points of intersection with the exterior line. Here is a detailed verification by enumeration:

Denote the points of the

GRAPHICAL DIFFERENCE SETS AND PROJECTIVE PLANES

219

exterior line as a, b, c, d, e, and the vertices of K20 (say in clockwise order) as 1,...,20.

The five bundles of five parallel lines, besides the

bundle first described, are the five columns of five lines, listed below.

16(11) (16)a 27(12) (17)b ' 38(13) (18): “(l/o) (19)d 5(10) (15) (20)'e

l34(10)b 578(1A)e 9(11) (12) (18)e (13) (15) (16)2d (17) (19) (20)6c

356(12)d 79(10) (16)c (11) (13) (14) (20)b (15) (17) (lBMs (19)128e

4.

245(11)c 689 (15)]: (10) (12)(13)(19)a (14) (16) (17) 3e (18) (20) 17d

467(13)e 8(10)(11)(17)d (12) (14) (15)1c (16) (15) (19)5b (20)239s

THE AFFINE PLANE AG(2,7) Regard K

49

as the join of K

7

and K

52

.

Then K

7

and the six equiskip—S

heptagons of K42 are one bundle of seven parallel lines.

The seven equi-

akip-6 hexagons determine one bundle of seven parallel lines which intersect the exterior line, K7, in its seven points.

Together the equiskip heptagona

and hexagans cover the edges of skips 5, 6, 11,

13, 17, 20 of K

The re—

42‘ maining forty-two lines, which also intersect K7, are determined by the fortytwo hexagans which have the remaining skips O, 1, 2, 14, 15, 16, 18,

19 of K42, as marked in Fig.

FIGURE 2

3, 4, 7, 8, 9, 10, 12,

2.

1 9 7

16 12

14 4

19521

8

a

‘ 5.

AFFINE PLANES 0F NONPRIME—PONER ORDER Until he discovered a slight arithmetical error, the author thought he

had a model for AG(2,10), analogous to the one above for AG(2_,7).

That is,

220

SOBCZVK

[(100 is regarded as the join of K10 and K90.

The nine equiskip-S decagons

of K90 and [(10 are one bundle of ten parallel lines.

The ten equiskip—9

nonagons of K90 determine another bundle of parallel lines which intersect the outer line, K

, in its ten points.

The equiskip decagons and nonagons

cover the edges ofoskips 8, 9, 17, 19, 26, 29, 35, 39, M.

The remaining

thirty—six skips of K90 should be covered by ninety rotational nonagons. ' 0n discovering his error, the author in haste assumed that if he could find

twelve series of rotational triangles to cover the skips, the triangles could be assembled into nonagons as needed, since there is a Steiner system on Ko' He did indeed find a profusion of solutions for twelve triples (closed triangles) of skips covering the thirty-six skips, but so far has not succeeded in assembling one of the solutions for the triples into a nonsgon.

Computer help

to decide this, and also to decide whether any of many other models for AG(2,10),€ AG(2,12), AG(2,15) proposed by the author are successful, has been promised by some of his colleagues who are good computer programmers. By the Bruck-Ryser theorem, there does not exist a finite plane of order six.

The edges of K43 are covered uniquely by the seven rotational series

of triangles with skips 0, 16, 17; 1, 10, 12; 2, 15, 18; 3, 4, 8; 5, 13, 19; 6, 7, 14; 9, 11, 20.

These cannot'be assembled into a heptagon, for if they

could be, we would have a model as in section 2 for PG(2,6).

0n the other

hand, the nonagons in the model of PG(2,8) are dissectible into twelve rotational series of triangles which uniquely cover the edges of K73.

The tri-

angles of course may be reassembled to form the nonagons. Marshall Hall has proved that there does not exist a cyclic‘ or rotational projective plane of order ten.

The existence of a rotational affine

plane with one exterior point in the lower order cases, in which cyclic prajective planes exist, suggests to the author that Hall's result may imply also that there does not exist a rotational model with an exterior point for an affine plane of order ten.

(Also, see [19].)

But perhaps there is, as in-

dicated above, a rotational affine model with an exterior line, exterior pair of lines, exterior pencil, or et cetera.

6.

DUAL PLANES The dual AG(2,n)* of AG(2,n) is PG(2,n) with deletion of one pencil and

its "focus," but not of the other points of the lines of the pencil.

In

AG(2,n)*, not every two points are on a line, but every two lines are on a point.

The number of lines per pencil is n (the same as the number of lines

GRAPHXCNL DIFFERENCE SETS AND PROJECTIVE PLANES

221

per bundle in AG(2,n)); the number of points on each line is (n + 1) (the same as the number of lines per pencil in AG(2,n)).

model for AG(2,3)* is the join K4 + K3:

For example, the obvious

the nine lines are K4, and the eight

skip 0, l, 2 triangles of K8 with their points of intersection in K4. Ex— plicitly, if 1, 2, 3, A are the vertices of K!" and 5, ..., 9, 0, n, B the vertices of K8, then the nine lines of AG(2,3)1 are 123A, 5681, 6792, 7803,

89M, 9081, 00.52, 0:563, 857/».

The twelve uncovered edges (pairs of points

which are not on lines) are the diagonals 59, 60,

bipartite edges 17, 28, 39, 40, lo, 28, 35, A6. the join K5 + K15.

7o, 85 of KS' and the eight

The model for AG(2,A)* is

The sixteen lines are K5 and the fifteen quadruples in

K15 with outer skips l, 2, 3, 5, inner skips 4, 6, together with their points

of intersection in KS.

The pairs of points which are not on lines include

the ends of each of the skip-0 edges. For AG(2,5)* = K6 + K24, the twenty—five lines are K6 and the twenty— four pentagona having outer skips 0, 1, 5, 10,

' with their points of intersection in K6.

3, inner skips 2,

7,

6, 8, A,

The uncovered edges of [(30 include

the twenty-four skip-9 and twelve skip-11 edges in K2,. For the above model of AG(2,5)*, the pentagons with outer skips 1, 5, 3,

8, 2,

inner skips lo,

7, 9, 10,

11, cannot serve as lines, because there

are only twelve skip-11 or diagonal edges in K24, so that the pairs of endpoints of the diagonals each would he on two lines.

These latter pentagons,

however, form some kind of partial combinatorial design on twenty—four ob— jects:

the pairs corresponding to the skip—0 and skip-6 edges are not cov-

ered; the pairs corresponding to skips-1, ..., 5, 7, ..., 10 are uniquely covered; and the pairs corresponding to the diagonals are each covered twice;

by the twenty-four pentagons or blocks of five objects.

REFERENCES 1.

A. A. A1bert.and R. Sandler, An Introduction to Finite Projective Planes, Holt, Rinehart and Winston, New York, 1968.

2.

P. Dembowski, Finite Geometries, Springer-Verlag, New York, 1968.

3.

J. w. P. Hirschfeld, Projectivs Geometries Over Finite Fields, Clarendon ' Press, Oxford, 1979.

4.

A. J. Hoffman, Cyclic Affine Planes, Can. J. of Math. 4(1952), 295-301.

5.

1). Frank Hsu, 'L‘yclic Neof‘ields and Combinatorial Designs, Springer-Verlag, New York, 1980.

6.

D. J. A. Welsh, Matraid Theory, Academic Press, New York, 1976.

222

SOBCZYK

NOTES ADDED IN PROOF Following is an easy matricial construction for AG(2,p), p prime, which shows the existence of a (graphical) model which is cyclic in a weaker sense

than that of [1.] and which does not require that there he an exterior point, or fixed point of a cyclic collineation. contrast with those in [4], or mixed nature.

See Remark at p. 300 of [In].

In

the difference sets involved here are of a partial

(The author poses as a problem to find a similar matricial

construction for the Galois AG(2,pk) k > 1.) For the graphical—cyclical model for AG(2,3), numerate the points of K9

as in Fig. 3.

The vertical bundle of three parallel lines consists of the

three equiskip-Z triangles, i.e.,

0

1

Z

10

ll

12

20

21

22

the rows of

The horizontal bundle I consists of the lines 10

20

ll

21

12

22

The next bundle II,

11

22

12

20

10

21

is obtained from I by taking the diagonal of I as first line, and counting cyclically to obtain two further lines. The last bundle III,

12

21

10

22

11

20

is obtained similarly from 11.

Referring to K9, the lines of bundle I are

three skip 0, 0, 1 triangles; the lines of bundle II are three skip 0. 3, 3

GRAPHICAL DIFFERENCE SETS AND PROJECTIVE PLANES

223

triangles; finally, the lines of III are three skip 1, 1, 3 triangles.

The

twelve triangles, in cycles of three, cover all of the skips 0, 1, 2, 3 E 6 of 159. FIGURE 3

21

2

To further convey the idea, which produces a quasicyclic model for each prime 1), consider AG(2,5).

Disposing of excess notation and punctuation,

bunt-‘0

bunt-IO

bump-Io

L‘WNHO

kwNHo

the horizontal bundle I is

with 11, III, IV, and V, respectively, 0

1

2

3

4

0

2

4

1

3

1

2

3

4

0

1

3

0

2

A

2

3

A

0

1

2

lo

1

3

0

3

4

0

1

2

3

0

2

la

1

lb

0

1

2

3

4

l

3

0

2

0

3

1

A

Z

0

lo

3

2

1

1

lo

2

0

3

1

0

4

3

2

2

0

3

1

4

2

1

0

A

3

3

1

lo

2

0

3

2

l

0

A

A

2

D

3

1

A

3

2

1

0

224

SOBCZYK

In K25, the vertical bundle consists of the five equiskip-A pentagons, which cover all edges of skips A, 9.

Bundle 1 consists of five 0, 0, 0, 0,

gons, which cover edges of skips 0, 1,

2, 3.

5, 5, 5, 5, pentagons, which cover edges of skips 0, 5, 6, 11. consists of five 2, 2,

10.

7, 2,

3 penta—

Bundle 11 consists of five 0, Bundle III

7 pentagons, which cover edges of skips 2, 5,

7,

Bundle IV consists of five 6, 1, 6, 6, l pentagons, which cover edges

of skips l, 6,

B, 10.

Bundle V consists of five 8,

which cover edges of skips 3, 7, 8, ll.

3, 3,

3, 3, pentagons,

Altogether, the thirty pentagons,

in six cycles of five, cover all of the skips 0, 1, 2, 3, A,

10, ll (twenty-five edges of each skip) of K25.

5, 6, 7, 8, 9,

‘

For a start on an AG(2,10), the five equiskip-h decogons (= complete sub— graphs K10)

of K50 may be taken as five vertical lines.

of skips 4, 9, 14, 19, 24.

These use the edge

A system of one hundred pentagons

(= complete

subgraphs K5), formed from the twenty remaining skips of 0 through 23, perhaps in twenty cycles of five pentagons each, would be equivalent to three mutually orthogonal Latin squares of side 10. an open research question.

Whether there are three such equates is

Mostly by experimental search (unaided by comput-

er), the author has found systems of one hundred edge—disjoint Ks's, but un-

fortunately for none found so far is the remaining subgraph of K50 the union (edgewise) of five Klo's.

Neither has the author been able to muster suffi-

cient number—theoretic resources for a proof of nonexistence (say in cycles of five), nor has the earlier-promised computer programming help from his colleagues been forthcoming. A system of one hundred pentagons, in cycles of ten, covering the edges of the twenty skips, is equivalent to four mutually orthogonal Latin squares. The author has produced systems of quadruples of K40, in cycles of five, but which give rise only to two orthogonal squares; cycles of ten would imply ten bundles of parallel 4—point segments and therefore three mutually orthogonal squares .

1

TILING THE PLANE HITH INCONGRUENT REGULAR POLYGDNS* Hans Herder?

Department of Applied Mathematics The Heisman Institute of Science Rehovat, Israel

,

Professor Michael Edelsteiu asked this author how to tile the Euclidean plane

with squares of integer side lengths all of which are incongtuent.

The ques-

tion can be answered in a way which involves an perfect square and a geometric application of the Fibonacci numbers. A perfect square is a square of integer side length which is tiled with more than one (but finitely many) component squares of integer side lengths all of which are incongruent. articles [3 and 5].

For more information, see the survey

A perfect square is simple if it contains no proper

subrectsngle formed from more than one component square ; otherwise, it is compound.

It is known ([3], pl. 884) that a compound perfect square must

have at least 22 components.

Duijvestijn's simple perfect square [2] (see

Fig. 1) thus has the least possible number of components (21).

*Presented by title . Wt affiliation: Boston, Massachusetts

Department of Mathematics, Boston State College,

HERDA

226 FIGURE 1.

27

35 50 8

19 15

. a

9

29

25

17

H

6

7

16

2

‘8

4

‘ 4

33

37

42

The Fibonacci numbers are defined recursively by El - 1, f2 = 1, and

fn+2 - in + fn+1

(n a 1)

They are used in connection with the tiling sham in Fig. 2.

(*) Its nucleus

is a Zl-component Duijvestijn square, indicated by shading in Fig. 2, having side length 3 = fl - s = 112, as in Fig. 1. FIGURE 2.

135

On top uf this square we tile a one—component square 3 of side length f2 - s = s = 112, forming an overall rectangle of dimensions 25 by 5.

0n

the left side of this rectangle (the longer edge) we tile 3 square 23 of side length f3 - s - Zs - 224, forming an overall rectangle of dimensions

35 by Zs.

We now proceed cuunterclockwise as shown, each time tiling}

TILING THE PLANE HITH POLYGONS

227

square fns onto the required longer edge of the last overall rectangle of

dimensions fns by fn_1s, forming a new overall rectangle of dimensions fn+1s by fns (this follows from (*)) .

The tiling can continue indefinitely in this

way at each stage because fns - fn_1s + fn_4s + fn_3s (this is used for n z 5

and also follows from (’0).

A closely related Fibonacci tiling for a single

quadrant of the plane (but beginning with two congruent squares) occurs in

([1], p. 305, Fig. 3). If we consider the center of the nuclear hatched square as the origin 0 of the plane, it is clear that the tiling eventually covers an arbitrary disc centered at 0 and thus covers the whole plane,

Finally, note that all the

component squares used in the tiling have integer side lengths and are incongruent. The tiling described above may be called static, since the tiles remain fixed where placed, and the outward growth occurs at the periphery. also interesting to consider a dynamic tiling. 'square. of 56.

It is

Start with a Duijvestijn

Its smallest component has side length 2.

Enlarge it by a factor

The smallest component in the resulting square has side length 112.

Replace it by. a Duijvestijn square. again by a factor of 56. the tiling.

Now enlarge the whole configuration

Repeat this process indefinitely, thus obtaining

Here no tile remains fixed , outward growth occurs everywhere,

and it is impossible to write down a sequence of side lengths of squares used in the tiling.

The three—dimensional version of this tiling problem (due to D. F. Daykin) is still unsolved:

can 3-space be filled with cubes, all with integer side

lengths, no two cubes being the same size?

([4], p. 11).

The plane can also be tiled with incongruent regular triangles and a

single regular hexagon, all having integer side lengths. FIGURE 3.

228

HERDA Begin with regular hexagon I (see Fig. 3) and tile regular triangles

Now tile .8 regular triangle with side length 7 along the sixth

side of the hexagon.

This counterclockwise tiling can be continued indef-

initely to cover the plane.

The recursion formula for the side lengths of

the triangles is:

s 1 -1

'

(1=1—5),

36=7,s1-s

1—1

+s

1-5

(127)

ACKNONL EDGMENT The author would like to thank the Weizmann Institute of Science in Rehovot, Israel, where this work was done.

REFERENCES 1.

A. Brousseau. Fibonacci numbers and geometry, The Fibonacci Quarterly 10(1972), 303-318,323.

2.

A. J. W. Duijvestijn. Simple perfect squared square of lowest order, J. of Combinatorial Theory (3) 25(1978), 555—558.

3.

N.‘D. Kazarinoff and R. Weitzenkamp. Ail. Math. Monthly 80(1973), 877-888.

4.

Problems in Discrete Geometry, collected and edited by William Maser with the help of participants of Discrete Geometry Week (July 1977, oberwolfach) and other correspondmts; 3rd ed., 1978, unpublished.

Squaring rectangles and squares,

5.

W. T. Tutte.

The quest of the perfect square, Am. Math. Monthly, 72

(1965), No. 2, Part II, 29-35.

A (.._. _.< m.

as shown.

_ A--- .._.._.___..._..t..i__._

with side lengths l, 2, 3, In, 5 (and so named) counterclockwise around it

PROBLEMS"

1.

(Steven Lay)

Let K be a convex subset of E“ and suppose that the

interior of K is nonempty.

Find necessary and sufficient conditions on K

for there to exist a nonconvex set S in IEu such that the kernel of S is equal to K.

In particular, does there exist a nonconvex subset of E2 whose kernel

is the closed unit ball?

(NOTE:

Victor Klee has pointed out that this prob-

lem was originally posed by Fejes—Toth and was essentially solved by Klee in his paper, A theorem on convex kernels, Mathemtika 12(1965), 89-93.

His

result applies to the case when K is a closed, convex set in a separable Banach space.

Marilyn Breen independently obtained essentially the same result

for Rd and also a result on nonclosed sets in her paper "Admiaaable Kernels for Starahaped Sets," Proc. of AMS, 82, No. A (1981). 622—628.

David Kay has

*Ed—tltorial Note: The first two problems stated above were the only onespresented at the Problem Session of the conference, the majority of the problems having been posed during the presentation of papers. Those problems appear in previous chapters of these Proceedings and will not be duplicated here. The other problems are the result of another convexity conference held only one month later (Special Session on Combinatorial Geometry and Convex sets, 777th Meeting of the American Mathematical Society, Davis, California,

April, 1980). The organizers of that conference, G. D. Chakerian and David Barnette, and the editors of these Proceedings have agreed to include those problems here as a service to the mathematical community.

-.

229

230

PROBLEMS

recently generalized Klee's result for noncomplete spaces in "Starshaped Sets

with Prescribed Convex Kernels in Separable, Normed Linear Spaces", submitted for publication.

The original problem remains for sets in the general setting

of linear topological spaces, and almost no results of a general nature have been obtained for the case when K is closed, even for E“).

2.

(David Kay)

Define a convex representatian of an alignment of‘x

to be a one-to-one convexity—preserving map from X to the relativized con-

vexity space of a subset of Rd for some d using the convex hull in Rd. representation is affine if we use the affine hull in Rd. cerning which almost nothing is known, are:

A

Two problems, con-

'

(a)

Characterize those alignments which have a convex representa-

(b)

Characterize those affine alignments (i.e., matroids) which

tion.

have an affine representation.

3.

(Richard Gardner)

Suppose K is a compact convex subset of the plane,

and three lines L1, L2, L3 meet at a point in K, dividing K into 6 regions, labelled cyclically by (xl,y2,x3,y1,x2,y3).

If IEI denotes the area of E,

it may be shown that

I‘ll PM

+

ll |y2|

+

1x31 |y3|

>~3

(1)

2

x + x 31 + x 3| I x 1 + x 2| > 3 I2 __ + __ +— I’ll

lyzl

(2)

lyal

(It suffices to take K to be a triangle.)

It is conjectured that equality

in (1) and (2) above holds if and only if K is a triangle and the lines L1, L2, L3 pass through the centroid of K, parallel to the sides of K.

(See

the article R. J. Gardner, 5. Kwapien and D. Laurie, Some inequalities related to compact convex sets, to be published.)

4.

(Jacob K. Goodman)

This problem was conveyed to me by R. Pollack,

who heard it from A. Lax, who originally got it from a draftsman:

If three

rays meet at a point, determine all the triangles having vertices on those three rays .

PROBLEMS

231

If the rays are mutually perpendicular, it is not hard to see that the

answer is:

all acute triangles.

D. Kay

(private communication) has gener—

alized this to the following partial solution:

angles 9, o, I, each > 11/2, then the answer is: a, B, y satisfy max(u,B,y) n 1' 2;

moreover, there are n-gons H for which K has area only arbitrarily little more than n E 2. 11'!

The question is:

How does one find such a polygon K inside

There is a (finite) algorithm described in the paper cited below, in the

case n < 5, for locating K (its sides cannot always be obtained by prolong— ing the sides of I”).

The problem then is to extend this to an algorithm

which works if n > 5.

(See J. E. Goodman, 0n the largest convex polygon con-

' tained in a nonconvex n—gon, or how to peel a potato, Geometriae Dedicata. to be published.) 6.

(Branko Griinbaum)

It is well known that, for each n > 3, the eucli-

dean plane Ez can be tiled by convex polygons each of which has n sides; thus each convex polygon is a conbimtorial prototile of some tiling of the plane.

In 1975 the following problem was communicated to me by Ludwig Danzer:

Is

every (bounded) convex polyhedron in E3 a combinatorial prototile of some

tiling in E3? 7.

(Branko Grfinbaum)

The situation in Problem 6 is drastically altered

if instead of "combinatorial prototiles" we consider "congruence prototiles." If a convex polygon is a congruence prototile it has at most 6 sides, but not every convex polygon with at most six sides is a congruence prototile (e.g., the regular pentagon). No characterization is known for those that are (see

D. Schattschneider,‘ Tiling the plane with congruent pentagons, Math. Mg. 51 (1978), 29—44). Determine the largest number of faces (or of vertices) pos— sible in a convex polyhedron which is a congruence prototile for Es. '(Note: The crystallographer Peter Engel from Bern recently found examples of such polyhedra with up to 38 faces and up to 70 vertices; see the account in B.

Grunbaum and G. C. Shephard, Tilings with congruent tiles, Bulletin of Amer.

mtha Soc. 3(1980), to be published.)

232

' 8.

PROBLEMS

(J. H. Wills)

Let Kg, Kg. K: be the three convex bodies which define

the three classical metrics:

Euclidean (ball), cube and octahedron and with

d dimensional volume WK?) = l, d = 2, 3,... with ”d the volume of the unit sphere.

Thus ,

K: _ “‘1'"”xa)/(E:=1 x12 1/2 ‘ ”El/d}

x; = {(x1,...,xd)/Zj=l 11%|“2 < (mm) x; ='t(x1....,xd>/n1u::1x"l.,d [x1] < 1/2} If KL = x: n K3], 1 < :1, then trivially o < v(1(:j)< 1.

(a) Do the three limits 1m v(x 0 or = 07 (G. L. Alexanderson and John E. Wetzel)

dimensional flats and C'(r)

Let fk be the number of kw

the number of bounded r-dimensional cells formed

by an arbitrary arrangement of hyperplanes in Ed, and consider the inequalities , r _ k d — k C(r)>2k_o(1) [k—r)fk

for 0 < r < d.

3. ()

In our paper, "Arrangements of planes in space," to appear

in Discrete Mathematics, we prove ('0 for all non-degenerate arrangements, 1.2., arrangements whose normals span Ed, in the cases d I 2 and d - 3; and

we determine the arrangements for which the equalities hold. Although the inequalities (*) seem to hold for a wide class of arrangements in E4, they do not hold for all non—degenerate 4—arrangements.

Pre-

cisely when are they true for d = lo, and what is the situation for d > A?

10.

(Paul Erdiis; posed jointly with Szemerédi)

points in the unit square (or unit circle).

Let x 1

, . . .,xn be u

Denote by D(x1,...,xn) the small—

est distance d(xi,xj), 1 < 1 (j < n and by o(x1....,xn) the smallest angle

e

,then determined by these n points. We conjecture that” D(x ----,x)> 1 n ln l . More generally, put n

«(x1,...,xn) = o

F(n) =

max (D(x1,...,xn)-a(x1,...,xn)) x1, . . . ,xn

PROBLEMS

233

Prove F(n) - o[ 31/2] and determine P(n) as well as you can. obvious.

No) > an"2 is

n

Background:

Let x1. ,,,.xn be n points in the unit square (or circle). n A('x1,...,xn) is the smallest area of all the [3] triangles (x1,xj,xl). Put

A(n) =

max A(xl,...,xn) xl' ... s xn

The determination of Mn) is known as Heilbrorm's problem.

Heilbronn claimed

he only transmitted it, but since he is unfortunately cured of our incurable disease we cannot find out.

It was conjectured that

c c l 2 —Z < A(n) < —2 n n

(1)

K. F. Roth first proved A(n) = o i). later Roth.

This was sharpened by W. Schmidt and

I observed A(n), > cl/nz.

Recently Komlde, Pintz and Szemerédi

proved

clogn

c

—3—— cn1+r_2, sharpening a previous result of Kirteszi which stated fr(n) > on log n.

(See S. A. Burr, B. Grunbaum and N. J.

A. Sloane, The orchard problem, Geometriae Dedicate: 2097!»), 397-524 and B. Brflnbaum, New Views of old questions of combinatorial geometry, Call. Internasianale Tear-115 Cambimtorie, Roma 1973, Accad. Nos. Limei, Tom I (1976), 161—468.)

15. line.

(Paul Erdfls)

that L1 has “i points, (:1 >

states that am - Z.

..,xn be n points in the plane not all on a

Let xl,.

Join every two of them.

a2

We then obtain the lines L1,...,Lm. >

> am.

Gallai'l theorem easily implies m > n.

the number of possible choices of (“133--,am}-

F(n) < a“

Assume

A classical theorem of Gallai

Denote by F(n)

I conjecture that

1/:

It is not hard to see that apart from the value of C this--if true-d: best possible.

(For further problems of this kind see P. Erdfis, Same combinatorial

problems in geometry, Lecture Notes in Math 792, Geometry and Differential Geometry, Prov... Haifa, Israel (1979), 46-53; this paper contains extensive references .)

AUTHOR INDEX

'

Ahler, I., 4

Duchét, P., 174

Alexander, R., 107 Alexandroff, P., 22 Altshculer, A., 11

Duijvestijn, A. J. W., Duke, 15, 16

Dudley, R., 166 225, 226, 227

Bajmficzy, E., 160 Banchoff, T. F., 13 BArSny, 1., 160 Barnetce, D., 1, 30, 164

Eckoff, 3., 151, 154, 159, 161, 162,

Beck, 8

Ehrhart, E., 52 Buns, T. C., 210, 213 Euler, L., 10

163, 165, 168, 174, 176 Edelman, H., 146 Edelstein, M, 225

Berge, C., 53, 174 Billera, L. J., 1, 4, 31 Birch, 3., 152, 155, 180 Bitkhoff, G. D., 85 Bjfirner, A., 35 Bland, R., 164

Falconer, K. J., 107

Fenchel, W., 58, 90 Firey, W., 64 Fujiwara, M., 63

Bleichet, 8 1501, G. 63

Boudeaen, AB., 39 Bonnesen, T.,

58,

Gierz, C., 146 Goodman, J. E., 135

90

Breeu, M., 163, 166

Gruber, P., 193 Grunbaum, 3., 1, 15, 16, 21, 32, 33, 115, 151, 167, 169, 173, 174, 182, 207, 210

Bruckner, M., 30

Brylawski, T., 164

Conn-Vossen, S., 103 Conway, 12 Cordovil, R., 80 Crofton, M., 167 Crow, 8

Hadwiger, H., 53, 68, 70, 71 Hajds, C., 203 Hall, M., 220

Hammer, P. C., 174 Hammer, R., 167 Hare, W., 162

Danczig. G. B. ,

,4

Danzer, L. , 115,1151, 167, 169, 173, 174

Day, A., 143 Dayktn, D. F.,

Helly, E., 173 Hilbert, D., 58, 103 Hirsch, W. M., 1 Hoffman, A. 1., Hopf, H., 22

227

146

Dehn, M., 23 De Smet, H. J. P., 120 Dirac, G. A., 119

Iversland, L., 157

Doignon, J. P., 119,151,153,154, 155,161,162,167,168,175, 178m, 179, 131, 183

Jacobi,

235

87

AUTHOR INDEX

Z36 Jamison-Waldner, R., 157, 167, 168, 174, 175, 176, 179

Rado, R., 146

Jucovic, E., 210

Reay, 1., 175, 179, 180. 183, 184

Radon, J.. 173 Relsner, G. A., 29

Robinson, 137 Kuhn, J., 146 Katchalski, 11., 132 Kay, D. C.. 167, 168, 174, 178

Robinson, R. 14., 204

Rota, G. (2., 49 Rnsnak, K., 16

Keller, 0., 203 Kennelly, J., 162 Kim, S., 10

K1ee,V., 22, 23, 32, 33, 115, 151, 167, 169, 173, 174 Kramer, 1)., 165 Krasnoselskii, M. A., 81 Kruskal, J. 13., 17

Scarf, H.

E., 119

Schlfifli, 23 Schmidt, J., 114, 116 Schneider, 11., 193

Sierksma, 6., "151, 155, 159, 167, 168

Simntis, 1., 10, 11, 13

Kung, J. P. 5., 146

Sinai, Ya. G., 86

Kuratorwski,

Shephard, G._ C., 1, 47

122

Las Vergnas, 14., 164

Shloeman, 160 Salmnerville, D. H. Y., 23 Stanely, R. P., 28, 29, 31, 32 Staude, 103

Lawson, .T. D., 146 Lee, (1., 1

Stein, 5. K., 204 Steiner, J., 73

Levi, 1., 168, 174

Steinitz, 13., 7, 15

Lindquist, N., 164

Stall, A., 89, 94

Lindstrfim, 3., 117

Strangeland, J., 158

Lockwood, E. IL, 86 £05, 137

Sylvester, 166

Mandel, A., 80 Mani, P., 4, 12, 31 Mankiewicz, P., 195

Todd, M. J., 4

Lamen, D. G., 4, 158, 161, 190

Mattheiss, T. 11., 33

McMullen, 17., l, 24, 27, 28, 31, 39, 52, 53, 161 Meier, Ch., 53

Szucs, A., 160

Tutte, W. T., 122

Tverberg, 11., 117, 152, 153, 156. 153, 175 ,

Minkawski, H., 58, 203

Valette, G., 151, 153, 154, 161, 162, 167 Van Maaren, 11., 120

Moore, E. H., 114, 116 Maser, L., 108 Motzkin, T. 5., 26

Walkup, D. 14., 4, 24, 31

Meir, A., 108 _

Weissback, 3., 61, 62

Willa, R., 117 Perron, 6., 203 Peterson, 11., 63, 151, 152, 181

Wilson, R., 119 Winder, 11., 164

Petty, C., 162

Womble, E. VL, 167, 168, 174, 178

Poincare, H., 23

Woodcock, M. 11., 63

Pollack, R., 135

Provan, J. 3., 4 Yanagihara, K., 64

PROBLEM INDEX

David Barnette, 17 René Foumeau, 202

John Reay, 152, 154, 156, 157, 158, 159, 163, 161», 165, 166, 169

Robert Jamison-Waldner, 145

G. Thomas Sallee, 53

Jim Lawrence, 20/» Erwin Lutwak, 110

Gerard Siexksma, 175, 176, 178, 180, 184, 191 Andrew Sobczyk, 222

237

SUBJECT INDEX

A

C

Abstract convexity, 120, 164, 167 Affine plane, 216, 219 matricial construction for, 222 rotational model for, 218

Alignment (also, Aligned space), 120, 167, 173, 174

category of alignments, 131

free, 128

, 123 140, 144

Antimatroid, 124, anti-exchange law of, variety of, 135

in abstract convexity, 167, 168,

176 in alignments, 122,

127

equal to unity, 140 and minimal forbidden subspaces,

143, 144 used to define varieties, 129, 137

affine (in Rd), 123 box, 130, 167, 176

ordinary (of Rd)

Carathéodory number:

125

Arrangement(s): of lines, 73 nonstretchable,

73-75 of pseudolines, 77-79 Axiomatic convexity, 120, 164, 167

B

Carathéodory's Theorem:

in abstract convexity, 118

in R2, 77 Category of alignments, 131 Cap body. 89 Caustic curves, 86

caustics, convex (see also Billiards):

1n R2, 86-98 in R3, 99-103

Chinese Remainder Theorem, 117 Circular disk,

characterization of,

63

Closure: ~a1gebraic closure system(s), 114, 115, 174 topological, 115, 116

Compact element: Barbier's Theorem, 59 Basis: positive, 3 of a variety, 14 2

193

Complex: Cohen—Macaulay, 29

Z—cell, 7

Billiards: bibliography for ,

104 convex caustic f or, 86, 98 Blaschke-Lebesgue Theorem, 61 Blaschke's relatio n, 60 Box alignment,

convex body, 57, element, 139

130

in Rd, 167, 176 Box, 1n R“, 204 Breadth, 127 Bruck—Ryser, theor

em of, 217 ‘

Constant width, 58, 63 Contraction, 132

Contractible varieties, 132 Convex: body, 57 caustic, for billiards, 86, 98 compact body, 88, 193 cone, generating, 194 subsets, of convexity structure, 120

Copoint, of matroids, 116, 128

240

SUBJECT INDEX

Core:

Fibonacci:

in abstract convexity, 137, 143,

144 m-core, 154,

167, 175-188

numbers,

226

tiling, 227 Finitary axiom (see also Domain finiteness), 115, 121, 136 Finite character, of classes, 122

D Dehn-Sommerville equations, 23, 49 and primitive Radon vectors, 166 and h—vectors, 25, 31 Dimension, in abstract convexity,

Finitely based (variety), 142 Folkman-Lawrence Theorem, 80 Forbidden minor, 140 Forbidden subspaces, minimal, 122, 123, 138-143

Function (see Embedding, map). 165

126, 135, 180, 183 Divisible:

m—, 152,.154 (m,k)—, 152, 155, 179, 184 Domain finiteness (see also Finitary axiom), 174

Downset, of order ideals, 124 Dual, duality:

of Kelly's Theorem, 76, 79 planes, 220,

221

E Eberhard's theorem, 209—210 Eckoff's conjecture, 184 Eckoff-Jamison relation, 168 Edge,

concave (convex), 14

Embedding (also Rulization), 211 in abstract convexity,

141

of polytopes into E“, 9, 13, 14 of compact convex subsets into normed linear space, 193 Envelope, of curves, related to convex caustics, 86

G Gale diagrams, 165 Gale transforms, 161, 163 Galois field, 216

General position, of points, 183 Graph, graph—theoretic, 7, 116, 118, 204, 206 bipartite, 204 block, 126 chordal, 118, 125, 142 chordless paths in, 118 2—connected, k—valent, 211, extreme point of, 119

213

geodesically convex, 118 monophonically convex, 118 oriented, 40

ptolemaic. 125 simple,

118

simplicia1, 118 tree, 126

Euler characteristic, 25 Euler's relation (equation):

in E3, 10, 11 general, 22, 23, 48-51 and h-vectors, 25

Euler—type relations, 45, 46 Extreme cluster, 187

Extreme point(s), 181, 187 of a graph, 119 Extreme set, 186

Extremely detachable, 124 Exchange law, matroids, 124 Exchange number, 168, 177

F

H Half—space(s) (see also Hemiapace): open, 181 Hausdorff metric, 71, 193

Height, of aligned space, 123 Kelly number, 167, 168, 173, 174 m—Helly number, 167, 175, 176, 177, 183, 184 (o,n)—He11y number, 184, 190 Helly's Theorem:

dual of, in R2, 76, 79 general,

in Rd, 175

in El, ‘77, 83 Facet splitting, 11n

Hemispace (see also Half-space), 125

SUBJEcT INDEX Hull, convex,

241 Lekkerkerker—Boland characterization (of interval graphs), 118, 122

75

of a family of subsets; 78 operator, properties of, 120

Hypercone , 123 Hyperplane, 118, 181

Levi enlargement lemma,

76

Levi's Theorem (or relation), 167, 174, 175, 185, 190 Linear programming, 1 Logic, first order logic, 137

I

Lower bound theorem, 30,

Incidence algebra,

31

50

Independence, (see also Basis, Ma— troids): algebraic, 152 full, 153 general position, 183 of a set, 127 strong, 152

M Map (see also Embedding): polyhedral, 9

semiaffine, 194, 199, 201 Mstroids, 124, 129, 140, 143, 144 copoint of, 116,

III-independent, 156, 183 Inductive, projective limits, 132 Irreducible: completely, 125 in ideals, 128

oriented, 79 Mazur—Ulsm theorem, 193, 198 Metric, Hausdorff, 71, 193 Minimal forbidden subspaces, 122,

123, 138, 142, 143

Isometry (see also Rigid motion), Isomorphism, between alignments

(see also Embedding). 121

128

exchange law of, 124

Minkowski-additive functions, 48 Minkowski decomposition, 60 Hinkowski sum, 57, 69, 70

Minknwski's conjecture, 203 Minor, 132 J

Moment curve, 26

Motzkin's conjecture, 27 Join, of collection of alignments,

Join-closed, variety, 131, 132, 141 Join—closure,

of e variety,

N

135 Normed linear space: - locally convex, 193-195, 199, 201 strictly convex, 194

K Ratchalski,

theorem of, 182, 183

Keller's conjecture, 203, 204, 206 Kirchherger's Theorem in R2, 77 Krasnoselskii's Theorem, 83 Krein—Milman property, 119, 122,

124 Krusksl, theorem of, 27

0 Order alignment, in partially ordered sets, 123, 126, 130 total ordering of, 124 monotone ordering of, 124, 135 Order convex (convexity), 123, 167 Order ideal, of monomials, 28-29,

L Larman—Helly (Radon) number, 191 Laman's function, 158 Lattice, 116, 140, 167 algebraic, 139 complete, 139

of product-closed varieties, 134 semilsttice, 167

124

p Parallel lines (bundles), 216 Partition (sea Radon partition) Perron's conjecture, 204, 206 Pivot,

feasible,

3

242

SUBJECT INDEX

Poincaré's formula, 107

Radon vector, 166, 189

Polygon, regular, 215 Polyhedron, convex, 21

primitive, 165, 166 Rank, 127, 129, 140, 143 Realization (see also Embedding). 9

Csészér, 8, 12, 14 neighborly, 12

Releaux triangle, 61

Rigid motion(s), 70, 109 Rubber sheet geometry, 115

pointed, 33

polyhedral map, 9 Polytope, convex, 21 cylic, 26 in abstract convexity,

121

S

neighborly, 39 simple, 22, 39, 40, 4B simplicial, 21

Schlegel diagram, 12 Semilattice, 125, 167

3—valent, 209

Seminorm, 201 Separation properties: in abstract convexity, 122, 125,

toroidalLS, 11, 14 Probability vector, 166 Product, of alignments, 130 convex—product space, 167, 176

product-closed alignments, 130, 132 product—closed varieties, 134

Projective limits, 132 Projective planes, 143, 217 triangulation of,

16

134, 178 and Radon's Theorem, 164

in R2, 75, 77, 7e k—Shell diagram, 211

Shellable, 25 Shor—Khachian algorithm, 4 Simple circuit, 211 Simplicial basis, 3 Splitting (see Facet—, Vertex-

splitting) Spherical convexities. 167

Q

Starlike,

Quermassintegral, 58, 59, 68

81

finitely starlike, 82, 83 Steiner system, 215 rotational, 217, 218

R

Steinitz's theorem, 210, 213 d-Step conjecture, 2

Radon equivalent, types, 162, 163 Radon number, 117, 167, 174 m-Radon number, 167, 176, 184 (I,n)—Radon number, 184, 185, 190 Radon partitions: in abstract convexity, 174 and m—divisibility, 154 and Gale transforms, 161 and k—independence, 156 Ill-partition, 157, 175, 179, 187

(m,k)—partition, 152, 159, 163, 164, 179 primitive, 162-166 and "stolen" points, 190 Radon's Theorem: in abstract convexity, 167—169,

174, 184 in R2, 77

in Rd, 151-153, 159 and Radon vectors, 165

and separation properties, 164

string angle. 91

String curve, 88 String length function; 89 Subalgehras, 139 Subspace, of alignment, 121, 122,

Sum (see also Vector sum), 130 —closed, 130, 132 of alignments, 177 Support function, 47

T Tietze's theorem, local convexity,

Tile, tiling, 203, 204

of 32, 225, 227 lattice, 203

perfect square, of a square, 225 Torus: ' triangulation of, 8, 11' minimal triangulation of, 15, 16

139

SUBJECT INDEX

243

Tree (see also Graph), 126 Triangular picture frame, 8 twisted, 10 Tverberg's theorem; 152, 157 t and k—independence, 158 and n-partitions, 175

U unimarphism, 194, 201, 202

wuk, 199

‘

Universal algebras, 138, 139 Universal variety, 122

Upper bound theorem, 39 Upward LUwenheim—Skolem Theorem,

143

Valuation, 46

motion invariant, 67 nonnegative, 67 monotonic, 68 rational homogeneous, 70 simply additive, 69, 70 translation invariant, 70 Variety: contractible, 132 of alignments, 121 in universal algebras, 138, 139

lattice-closed, 140, 151 universal, 122

Vector sum (see also Minkowski decomposition), 57, 69, 70 f-Vector (of a polyhedron), 21 h-Vector (of a polyhedron), 24 Vertex: convex, nonconvex, 13

V

figure, 13 splitting, 15, 16, 17

LL

Valence (in space), 128

alumni-hook...

This book brings together twenty original papers on convexity and combinatorics for the benefit of graduate students and mathematicians in the fields of combinatorics, geometry, and analysis. Each paper is self contained and clearly written to enable the mature reader to easily understand the most current research in the field. The contributors to Convexity and Related Combinatorial Geometry are all recognized

experts and they report on important findings and up-to-date results. Highlights include an article by Victor Klee on the famous d-step conjecture which is pertinent to linear programming, and an article by Carl Lee on the recently solved problem of characterizing the f-vectors of a simplicial polytope.

The wide variety of research problems found in the papers and at the end of the book are excellent sources for “ I, 'inthe field. " L in im- andv ’ should take immediate advantage of Convexity and Related Combinatorial Geometry. about the editor! . . .

DAVID C. KAY is Professor of Mathematics at the University of Oklahoma. He received his MS. degree (1959) from the University of Pittsburgh and his PhD. degree (1963) from Michigan State University, East Lansing. Professor Kay has written numerous articles on geometry and convexity theory in mathematics and engineering journals, including one in the latest edition of Encyclopedia of Americana The author of an undergrad-

uate geometry textbook, Professor Kay's research interests are metric geometry, abstract convexity, linear programming, graph theory, and matroids. He is a member of the American Mathematical Society and the Mathematical Association of America. MARILYN BREEN is Associate Professor of Mathematics at the University of Oklahoma. She received her B.A. degree (1966) from Agnes Scott College and her PhD. degree (1970) from Clemson University. Professor Breen’s research has been in the area of convexity and her current work concerns combinatorial invariants of convex sets, including m-convex sets and starshaped sets. Professor Breen is a member of the American Mathematical Society and the Mathematical Association of America. Printed in the United State: ofAmerica

ISBN: 0—8247—1273-1

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