First Yaroslavl Summer School on Discrete and Computational Geometry. July - August, 2012

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P. G. Demidov Yaroslavl State University International B. N. Delaunay Laboratory Discrete and Computational Geometry

First Yaroslavl Summer School on Discrete and Computational Geometry July – August, 2012 Lecture Notes

Yaroslavl 2012

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UDC 514 First Yaroslavl Summer School on Discrete and Computational Geometry. July – August, 2012. Lecture Notes. — Yaroslavl State University, 2012. — 197 p. Funded by Russian Government Grant 220 / Contract 11.G34.31.0053

Delaunay Library Series Head Editors: N. Dolbilin, H. Edelsbrunner, A. Ivanov Editors: V. Buchstaber, V. Dolnikov, R. Karasev, V. Manturov, N. Moshchevitin, O. Musin, M. Nevskii, I. Sabitov, M. Schtogrin

Cover design by Xixi Edelsbrunner

ISBN 978-5-8397-0945-4

P.G. Demidov Yaroslavl State University

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Delaunay Library

The International Delaunay Laboratory of Discrete and Computational Geometry introduces a new books series: the Delaunay Library. Its mission is the advancement of research and education in this field. Specifically, we propose to publish scientific monographs, collections of papers on current research, and textbooks whose primary topics are in the fields of discrete geometry, computational geometry, and computational topology. The Delaunay Laboratory is being funded by the Russian Government since 2011 and is lead by Herbert Edelsbrunner. It constitutes a concerted effort of leading international specialists to create a new world-class research and educational center in Discrete and Computational Geometry on the premises of the P. Demidov Yaroslavl State University. The introduction of the Delaunay Library is part of its educational and promotional activities. The Delaunay Laboratory entrusts the Editorial Board of the Delaunay Library – consisting of three Head Editors, N. Dolbilin, H. Edelsbrunner, and A. Ivanov, and nine Editors, V. Buchstaber, V. Dolnikov, R. Karasev, V. Manturov, N. Moshchevitin, O. Musin, M. Nevskii, I. Sabitov, and M. Schtogrin – with the development of the library. For a book to be included in this series, it must be approved by this board. Additional information on the series, including annotations of the books in preparation, can be found on the web page of the Laboratory (www.dcglab.uniyar.ac.ru). Similar to the Laboratory, the series is named in honor of Boris Delaunay, a brilliant Russian geometer and one of the founders of modern discrete geometry. He was also a wonderful teacher and famous mountain climber. The Delaunay peak, more than four thousand meters high in the Altai mountains of Russia, is as well-known to alpinists as is the Delaunay triangulation to mathematicians. A stylized design of the peak created by Xixi Edelsbrunner can be seen on the front covers of the books in the series, and a hint of 3

Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»

the peak’s contour is given by the letter Λ in the Laboratory logo designed by A. Akopyan and S. Sharov–Delaunay. We open the series with the First Yaroslavl Summer School on Discrete and Computational Geometry, which contains lecture notes accompanying the short courses delivered in July and August of 2012. We hope this volume will be useful to all students and post graduates interested in the current state-of-the-art in this field. Currently, several additional volumes are in preparation. Among them is the Delaunay Volume, a collection of works by Boris Delaunay, selected and commented by his students N. Dolbilin and M. Schtogrin; the Voronoi Volume, a collection of papers by Georgii Voronoi, selected and commented by O. Musin; and a Russian translation of the textbook on Computational Topology by H. Edelsbrunner and J. L. Harer. We are confident that the Delaunay Library becomes a valuable resource for getting acquainted with, studying, teaching, and doing research in modern discrete and computational geometry and related topics. N. Dolbilin, H. Edelsbrunner, and A. Ivanov

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Table of Contents

N. Dolbilin, H. Edelsbrunner, A. Ivanov, O. Musin The First Yaroslavl Summer School on Discrete and Computational Geometry . . . . . . . . . . .

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В. Д. Володин Комбинаторика простых многогранников . . . . . .

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V. L. Dolnikov On Kneser’s problems . . . . . . . . . . . . . .

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A. O. Ivanov and A. A. Tuzhilin Optimal Networks . . . . . . . . . . . . . . .

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R. Karasev The Borsuk–Ulam and ham sandwich theorems . . . . . 111 M. Kerber Robust Geometric Computation . . . . . . . . . . 120 С. В. Матвеев Компьютерное табулирование трехмерных многообразий 147 Oleg R. Musin Kissing Problem in three and four dimensions . . . . . 164

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The First Yaroslavl Summer School on Discrete and Computational Geometry 1 N. Dolbilin, H. Edelsbrunner, A. Ivanov, O. Musin

We summarize the results of the First Yaroslavl Summer School on Discrete and Computational Geometry and asses its future perspectives.

The First Yaroslavl Summer School on Discrete and Computational Geometry was organized by the International Delaunay Laboratory “Discrete and Computational Geometry” and held in July and August of 2012 in a neighborhood of Yaroslavl. The aim of this article is to summarize the results of the School and to asses its future perspectives.

School on Discrete and Computational Geometry The 21st century rises as the era of science-based technologies and innovations. The field of Discrete and Computational Geometry provides some of the main tools for several application areas, ranging from entertainment to manufacturing, from medicine to facility planning, and more. Today, the field represents one of the most rapidly developing, perspective, and popular directions of modern mathematics and science. The Government of the Russian Federation supports the creation of a laboratory of discrete and computational geometry at the P.G. Demidov Yaroslavl State University under the leadership of Herbert Edelsbrunner, a pioneer in computational geometry and topology with commercial experience in the field. In 2011, the International Delaunay Laboratory “Discrete and Computational Geometry” was established. At the laboratory, specialists from Russia (Demidov Yaroslavl State University, Lomonosov Moscow State University, Saint Peterburg State University, Steklov 1 The paper was originally published in Modeling and analysis of information systems, Vol. 19, No 4, 2012. P. 168—173.

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Mathematical Institute), Austria (Institute of Science and Technology, Vienna), and the USA (University of Texas in Brownsville), join forces in research and education aimed at creating a modern word-class center of Discrete and Computational Geometry. An important activity at the laboratory is the teaching of graduate and post-graduate students, with the purpose of attracting young scientists to the field and training the future personnel of the laboratory. The Summer School is a significant part of this educational programme.

Goals and Objectives Mathematical Schools organized during winter and summer vacations have a long traditions in Russia. The main goal of the Summer School on Discrete and Computational Geometry is to improve the academic level of the graduate students in the subject. For this reason, the Delaunay Laboratory aims at a good compromise between the traditions of the Krein Winter Voronezh Mathematical School, organized since 1967 and oriented toward post-graduate and post doctoral levels, and that of the Summer School “Contemporary Mathematics”, organized since 2001 in Dubna and featuring introductory courses on a wide spectrum of modern mathematical topics. Several elementary courses were given at the school, by members of the Delaunay Laboratory as well as by invited professors from Austria, Japan, Russia, and the USA. These courses were supported by practical exercises conducted by graduate and post-graduate students with experience in the subject. In addition, a scientific seminar was organized, in which resent results in Discrete and Computational Geometry were presented and open problems were discussed. The informal contacts of the young mathematicians with each other and with older colleagues are of great value. New ideas can appear not only in the class room, but also during a volleyball game, while swimming in the Volga river, or during a table-tennis game. The main goal of the school is to provide the participants with plenty of food for thought.

Education and Program Below we give the complete list of lecture courses given at the School. We make no distinction between computational geometry, discrete geometry,

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and computational topology, since such divisions are not helpful. After all, in reality the best work results from interactions between pure science and applications, and this interaction can be complex and multi-facetted. One of the aims of the school was to show that such interdisciplinary cooperation can form a basis for progress in all the branches. Usually we had three 1-hour lectures and one seminar talk per day. The list of courses given at the School is as follows: • H. Edelsbrunner, Introduction to Computational Geometry and Topology, 4 lectures, see [1, 2, 3]; • N. Dolbilin, Introduction to the Theory of Periodic Coverings and Tilings, 2 lectures, see [4, 5, 6]; • O. Musin, Sphere Packings, 3 lectures; • S. Matveev, Computer Classification of 3-dimensional Manifolds, 4 lectures, see [7, 8]; • A. Ivanov, Optimal Networks, 3 lectures, see [9, 10, 11]; • I. Pak, Triangulations of Convex Polyhedra, 3 lectures, see [12]; • M. Myachin, Geometric Problems in Endoscopy, 3 lectures; • V. Volodin, Combinatorics of Simple Polytopes, 3 lectures; • V. Bondarenko, Graphs of Polytopes and Polytopes of Problems, 3 lectures; • V. Dolnikov, Kneser’s Problem and its Generalizations, 3 lectures; • R. Karasev, Borsuk-Ulam Theorem and Measure Partitions, 3 lectures, [13]; • M. Kerber, Robust Geometric Computation, 4 lectures, [14, 15], [16], [18, 19, 20], [21, 22, 23]; • N. Andreev, Mathematical Etudes, 2 lectures, see www.etudes.ru. The videos of the lectures are available on the site of the Delaunay Laboratory (http://dcglab.uniyar.ac.ru). In addition, a volume with lecture notes

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is under preparation and will appear in the Demidov Yaroslavl State University Publishing house. In addition to the lectures, some of the seminar talks are also available as videos on the same site. The list of seminars on new results and open problems is as follows: • H. Edelsbrunner, Which Convex Bodies Are Most Chiral? • O. Musin, The Thirteen Spheres Problem. • T. Samsonov, Geometric Methods in Cartographic Generalization. • N. Dolbilin, Local and Global Approaches to the Crystallographic Structure. • G. Ivanov, Deviation of the Convex Hull. • N. Strelkova, Minimal Networks on Convex Polyhedra. • B. T. Fasy, Metrics on Persistence Diagrams. • A. Magazinov, Voronoi Conjecture for a Special Class of Parallelohedra. • B. Wang, Elevation, Protein Docking and Gauss Map. • G. Chelnokov, Some Combinatorial Inequalities About Defining a Periodic Word by a System of Forbidden Subwords. • V. Fokin, Stability of Voronoi Diagram in Euclidean Plane. • V. Manturov, Graph-valued Quantum Invariants of Virtual Knots. • E. Bannai, On Spherical Designs. • J. Phillips, Computational Geometry on Uncertain Data. The School was concluded by a lecture on “The Discrete and Computational Project” given by H. Edelsbrunner, in which he talked about his understanding of the interaction between Mathematics, Computations, and Applications, illustrating the relationships with the development of a surface reconstruction algorithm (Computations), based on idea of discrete flow (Mathematics), with industrial uses in manufacturing and medicine (Applications).

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Concluding Remarks During the 19 days of the Summer School, 13 courses with a total of 40 lectures and 14 seminar talks were delivered to 50 students. During the practical exercises, most of the students passed a test on Computational Geometry and Topology. We therefore believe that the academic level of the participants has significantly improved. Very importantly, new communication channels have been opened and cooperations between participants have been initialized. All this gives us reason to conclude that the School has been successful. It is now time to start planning the Second Yaroslavl Summer School on Discrete and Computational Geometry that will be held by the International Delaunay Laboratory in the Summer of 2013. The information concerning the Second School will appear on the site of the Laboratory (http://dcglab.uniyar.ac.ru). Before closing this article, we would like to express our thanks to the Local Organizing Committee, and in particular to A. Yu. Uhalov, A. N. Maksimenko, and A. I. Garber, who took on the responsibility for planning and scheduling the School. Without their hard work, this School would have been impossible. We hope that the Summer School on Discrete and Computational Geometry in Yaroslavl will become tradition and an appreciated event within the wider Mathematical community.

References [1] Edelsbrunner H. Geometry and Topology for Mesh Generation. Cambridge Univ. Press, Cambridge, England, 2001. [2] Matveev S. V. Lectures on Algebraic Topology. European Math. Soc., Zuerich, Switzerland, 2006. [3] Edelsbrunner H., Harer J. Computational Topology: An Introduction. Amer. Math. Soc., Providence, Rhode Island, 2010. [4] Dolbilin N. P. Properties of Faces of Parallelohedra // Proc. Steklov Inst. Math. 2009. 266. P. 105–119. [5] Delone B. N. Geometry of positive quadratic forms // Uspekhi Matem. Nauk. 1937. No 3. P. 16–62 [in Russian].

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[6] Alexandrov A. D. Convex Polyhedra. Berlin, Heidelberg, New York, Springer, 2005 [Translation from: Moscow, GTTL, 1950]. [7] Matveev S. V. Algorithmic Topology and Classification of 3manifolds. Springer ACM-monographs, 2003; Moscow, MTsMNO, 2007. [8] Matveev S. V. Roots and decompositions of topological 3D objects // Uspekhi matem. Nauk. 2012. 67: 3(405). P. 63—114 [in Russian, English Translation: Russian Math. Surveys, to appear]. [9] Ivanov A. O. and Tuzhilin A. A. Branching Solutions to OneDimensional Variational Problems World Scientific, Singapore, 2000. [10] Ivanov A. O., Tuzhilin A. A. Extreme Networks Theory. Moscow– Izhevsk, Inst. Komp. Issl., 2003 [in Russian]. [11] Ivanov A. O., Tuzhilin A. A. One-dimensional Gromov minimal filling, arXiv:1101.0106v2 [math.MG] (http://arxiv.org), Matem. Sbornik. 2012. 203. P. 65–118. [12] Pak I. Lectures on Discrete and Polyhedral Geometry: Manuscript (http://www.math.ucla.edu/~pak/book.htm). [13] Kaplan H., Matouˇsek J., Sharir M. Simple proofs of classical theorems in discrete geometry via the Guth–Katz polynomial partitioning technique // Discrete Comput. Geom. 2012. 48. P. 499– 517. [14] Kettner L., Mehlhorn K., Pion S., Schirra S., Yap C. Classroom examples of robustness problems in geometric computations // Computational Geometry: Theory and Applications. 2008. 40. P. 61– 78. [15] Mehlhorn K., Yap C. Robust Geometric Computation: Manuscript (http://cs.nyu.edu/yap/book/egc/). [16] Liotta G., Preparata F. P., Tamassia R. Robust proximity queries: An illustration of degree-driven algorithmic design // SIAM Journal on Computing. 1999. 28. P. 864–889.

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[17] Burnikel C., Funke S., Seel M. Exact geometric computation using cascading // International Journal of Computational Geometry and Applications. 2001. 11. P. 245–266. [18] Funke S., Klein C., Mehlhorn K., Schmitt S. Controlled perturbation for Delaunay triangulations // Symposium on Discrete Algorithms. 2005. P. 1047–1056, [19] Halperin D., Leiserowitz E. Controlled perturbation for arrangements of circles // Symposium on Computational Geometry. 2003. P. 264–273. [20] Mehlhorn K., Osbild R., Sagraloff M. A general approach to the analysis of controlled perturbation algorithms // Computational Geometry: Theory and Applications. 2011. 44. P. 507–528. [21] Br¨ onimann H., Burnikel C., Pion S. Interval arithmetic yields efficient dynamic filters for computational geometry // Discrete Applied Mathematics. 2001. 109. P. 25–47. [22] Devillers O., Pion S. Efficient exact geometric predicates for Delaunay triangulations // 5th Workshop on Algorithm Engineering and Experiments (ALENEX), 2003. [23] Kerber M. Geometric Algorithms for Algebraic Curves and Surfaces, PhD Thesis, Saarland University, 2009 (http://www.mpi-inf.mpg.de/~mkerber/kerber_diss.pdf).

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Комбинаторика простых многогранников В. Д. Володин

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Выпуклые многогранники

Определение 1. Выпуклым многогранником называется выпуклая оболочка конечного набора точек в некотором пространстве Rn . Определение 2. Выпуклым полиэдром называется пересечение P конечного набора полупространств в некотором пространстве Rn : P = {x ∈ Rn : hli , xi 6 ai , i = 1, . . . , m},

(1)

где li ∈ (Rn )∗ - некоторые линейные функции, ai ∈ R, i = 1, . . . , m. Ограниченный выпуклый полиэдр называется выпуклым многогранником. Два предыдущих определения эквивалентны, т.е. подмножество в Rn является выпуклой оболочкой конечного числа точек тогда и только тогда, когда оно ограничено и является конечным пересечением полупространств. Размерность многогранника - это размерность его аффинной оболочки. Далее мы будем предполагать, что n-мерный многогранник (n-многогранник) содержится в n-мерном объемлющем пространстве Rn . Аффинная гиперплоскость H, пересекающая многогранник P n называется несущей (или опорной) гиперплоскостью, если многогранник целиком содержится в одном из определяемых ей замкнутых полупространств. В этом случае пересечение P n ∩ H называется гранью многогранника. Сам многогранник также считается гранью; остальные его грани называются собственными. Объединение всех собственных граней называется границей многогранника ∂P n . Каждая грань n-многогранника в свою очередь является многогранником размерности не выше n. Нульмерные грани называются вершинами, одномерные грани - ребрами, а грани коразмерности один гипергранями. 14

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Два многогранника P1 ⊂ Rn1 и P2 ⊂ Rn2 одной размерности называются аффинно эквивалентными (или аффинно изоморфными), если существует аффинное отображение Rn1 → Rn2 , устанавливающее взаимно-однозначное соответствие между точками этих многогранников. Два многогранника называются комбинаторно эквивалентными, если имеется взаимно-однозначное соответствие между их множествами граней, сохраняющее отношение включения. Заметим, что любые два аффинно изоморфных многогранника комбинаторно эквивалентны, но не наоборот. Более формальное определение комбинаторной эквивалентности использует понятия частично упорядоченного множества и решётки. Напомним, что частично упорядоченным множеством называется множество S с отношением частичного порядка 6 которое рефлексивно (x 6 x для всех x ∈ S), транзитивно (x 6 y и y 6 z влечёт x 6 z) и антисимметрично (x 6 y и y 6 x влечёт x = y). Далее все частично упорядоченные множества будут предполагаться конечными. Если отношение 6 применимо к любой паре x, y, то S называется вполне упорядоченным множеством. Любое вполне упорядоченное подмножество в частично упорядоченном множестве S называется цепью. Грани многогранника P всех размерностей относительно вложения образуют частично упорядоченное множество. Теперь можно заметить, что два многогранника комбинаторно эквивалентны тогда и только тогда, когда их множества граней изоморфны как частично упорядоченные множества. Определение 3. Под комбинаторным многогранником мы будем понимать класс комбинаторно эквивалентных выпуклых (или геометрических ) многогранников. Пример (симплекс и куб). Симплексом ∆n размерности n называется выпуклая оболочка набора из (n + 1) точек в Rn , не лежащих в одной аффинной гиперплоскости. Все грани n-симплекса являются симплексами размерности 6 n. Любые два n-симплекса аффинно эквивалентны. Стандартным n-симплексом называется выпуклая оболочка точек (1, 0, . . . , 0), (0, 1, . . . , 0), . . . , (0, . . . , 0, 1) и (0, . . . , 0) в Rn . Эквивалентно, стандартный n-симплекс задаётся (n + 1) неравенствами xi > 0, i = 1, . . . , n,

и x1 + . . . + xn 6 1.

(2)

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Правильным n-симплексом называется выпуклая оболочка точек (1, 0, . . . , 0), (0, 1, . . . , 0), . . . , (0, . . . , 0, 1) в Rn+1 . Стандартным q-кубом называется выпуклый многогранник I q ⊂ Rq , задаваемый как I q = {(y1 , . . . , yq ) ∈ Rq : 0 6 yi 6 1, i = 1, . . . , q}.

(3)

Эквивалентно, стандартный q-куб есть выпуклая оболочка 2q точек в Rq , каждая из координат которых есть 0 или 1. Два различных определения выпуклых многогранников приводят к двум различным понятиям многогранников общего положения. Набор из m > n точек Rn находится в общем положении, если никакие n + 1 из них не лежат на одной аффинной гиперплоскости. Тогда, с точки зрения определения 1, выпуклый многогранник является многогранником общего положения, если он является выпуклой оболочкой набора точек в общем положении. Все собственные грани такого многогранника являются симплексами, то есть любая гипергрань имеет минимальное возможное число вершин (а именно, n). Такие многогранники называются симплициальными. С другой стороны, скажем, что набор из m > n гиперплоскостей hl i , x i = ai , l i ∈ (Rn )∗ , x ∈ Rn , bi ∈ R (i = 1, . . . , m) находится в общем положении, если никакая точка не содержится более чем в n гиперплоскостях из этого набора. С точки зрения определения 2, выпуклый многогранник P n является многогранником общего положения, если ограничивающие его гиперплоскости находятся в общем положении. В каждой вершине такого многогранника P n сходится в точности n гиперграней. Многогранники с таким свойством называются простыми. Заметим, что каждая грань простого многогранника есть снова простой многогранник. Определение 4. Для любого выпуклого многогранника P ⊂ Rn определим его полярное множество P ∗ ⊂ (Rn )∗ как P ∗ = {l ∈ (Rn )∗ : hl , x i 6 1 для всех x ∈ P }. В выпуклой геометрии хорошо известно, что полярное множество P ∗ является выпуклым полиэдром в двойственном пространстве

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(Rn )∗ . Более того, если многогранник P содержит 0 в своей внутренности, то P ∗ является выпуклым многогранником (т.е. ограничен) и (P ∗ )∗ = P , см., например, [Zi, §2.3]. Многогранник P ∗ называется полярным (или двойственным) к P . Частично упорядоченное множество граней многогранника P ∗ получается из множества граней многогранника P обращением отношения порядка. В частности, если P — простой многогранник, то P ∗ — симплициальный, и наоборот. Пример. Любой многоугольник (2-многогранник) является одновременно простым и симплициальным. В размерностях больше 2 единственным многогранником, обладающим этим свойством, является симплекс. Куб является простым многогранником. Полярным многогранником для симплекса снова является симплекс. Полярным многогранником для трехмерного куба является октаэдр. Для любых двух простых многогранников P1 и P2 , их произведение P1 × P2 снова является простым многогранником.

2

Смежностные многогранники

Определение 5. Симплициальный многогранник S называется kсмежностным, если любые k его вершин порождают некоторую грань. Аналогично, простой многогранник P называется двойственно k-смежностным, если любые k его гиперграней имеют непустое пересечение (которое в этом случае является гранью коразмерности k). Очевидно, любой симплициальный (простой) многогранник является 1-смежностным (двойственно 1-смежностным). Можно показать, что если S является k-смежностным симплициальным n  многогранником и k > n2 , то S есть n-симплекс. Отсюда вытекает, что любой 2-смежностный симплициальный 3-многогранник есть симплекс. В то же время, существуют симплициальные n-многогранники с произвольным числом вершин, которые являются n -смежностными. Такие многогранники называются смежност2 ными. В частности, существует симплициальный 4-многогранник, отличный от 4-симплекса, любые две вершины которого соединены ребром. Пример (смежностный 4-многогранник). Пусть P = ∆2 × ∆2 — произведение двух треугольников. Тогда P есть простой многогран-

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В. Д. Володин

ник, и легко видеть, что любые две его гиперграни пересекаются по некоторой 2-грани. Таким образом, P является двойственно 2смежностным. Полярный многогранник P ∗ является смежностным симплициальным 4-многогранником. В общем случае, если простой многогранник P1 является двойственно k1 -смежностным, а P2 — двойственно k2 -смежностным, то их произведение P1 × P2 является min(k1 , k2 )-смежностным простым многогранником. Отсюда следует, что многогранники (∆n × ∆n )∗ и (∆n × ∆n+1 )∗ дают примеры смежностных симплициальных 2n- и (2n + 1)-многогранников. В следующем примере описывается смежностный многогранник с произвольным числом вершин. Пример (циклические многогранники). Кривая моментов в Rn задаётся как образ отображения x : R −→ Rn ,

t 7→ x (t) = (t, t2 , . . . , tn ) ∈ Rn .

Для каждого m > n определим циклический многогранник C n (t1 , . . . , tm ) как выпуклую оболочку m различных точек x (ti ), t1 < t2 < . . . < tm , на кривой моментов. Теорема 6. а) Циклический многогранник C n (t1 , . . . , tm ) является симплициальным n-многогранником; б) C n (t1 , . . . , tm ) имеет в точности m вершин x(ti ), i = 1, . . . , m; в) комбинаторный тип циклического многогранника не зависит от выбора t1 , . . . , tm ; г) C n (t1 , . . . , tm ) является смежностным многогранником. Доказательство. Рассмотрим известное тождество для определите-

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Комбинаторика простых многогранников

ля Вандермонда:  1 1 det x (t0 ) x (t1 )  1 1  t0 t1   .. .. = det  .  n−1 . n−1  t t1 0 tn0 tn1

19

 1 = x (tn )  ... 1  . . . tn  Y  . . .. (tj − ti ). = . .  n−1  06i 0 так что ti < ti + ε < ti+1 при всех i < m и выберем некоторое N > tm + ε. Определим линейную функцию Fτ (x ) как
det x , x (ti1 ), x (ti1 +ε), . . . , x (tik ), x (tik +ε), x (N +1), . . . , x (N +n−2k) . Эта линейная функция обращается в нуль в точках x (ti ) с i ∈ τ . В то же время, Fτ (x (t)) является многочленом от t степени n, который имеет n различных корней ti1 , ti1 + ε, . . . , tik , tik + ε, N + 1, . . . , N + n − 2k. Если i, j ∈ / τ , то между значениями t = ti и t = tj всегда имеется чётное число корней этого многочлена, так как вместе с корнем вида t = tl всегда имеется корень вида t = tl +ε. Следовательно, линейная функция Fτ (x ) имеет один и тот же знак во всех точках x (ti ) для i ∈ / τ , что означает, что τ соответствует набору вершин некоторой грани. Далее мы будем обозначать комбинаторный циклический n-многогранник с m вершинами через C n (m).

3

Перечисляющие полиномы

Пусть P n - n-многогранник. Обозначим через fi - количество его iмерных граней. Тогда вектор (f0 , . . . , fn ) называется его f -вектором. Полином f (P )(t) = f0 + f1 t + · · · + fn tn

(4)

называется f -полиномом. Полином h(P )(t) = f (P )(t − 1) = h0 + h1 t + · · · + hn tn

(5)

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называется h-полиномом. Коэффициенты fi и hi связаны соотношениями X i hk = (−1)i fi , (6) k i X i hi . (7) fk = k i В частности, h0 = f0 − f1 + · · · + (−1)i f i + · · · + (−1)n f n , является Эйлеровой характеристикой P n . Следовательно, h0 = 1. Для произвольного выпуклого многогранника это единственное линейное соотношение на числа граней, однако для простых многогранников существует [n/2] независимых линейных соотношений. Пример. Два различным комбинаторных простых многогранника могут иметь одинаковые f -векторы. Например, пусть P13 — трёхмерный куб, а P23 — простой 3-многогранник с двумя треугольными, двумя четырёхугольными и двумя пятиугольными гранями, см. рис. 1. (Заметим, что P23 является двойственным к циклическому многограннику C 3 (6) из определения 2.) Тогда f (P13 ) = f (P23 ) = (12, 8, 6). HH  HH   HH      H Z H  Z    H Z  Z  H    Z  @  Z @   @  @   @  @

P   @PPP P   Z @ Z   Z Z  @   Z  @  Z @   @  @   @  @

Рис. 1: Два комбинаторно неэквивалентных простых многогранника с одинаковыми f -векторами.

Теорема 7 (Соотношения Дена-Соммервиля). Для простого многогранника hk = hd−k

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В. Д. Володин

Доказательство. Пусть P n ⊂ Rn – простой многогранник. Выберем линейную функцию ϕ : Rn → R, принимающую различные значения на всех вершинах P n . Для этой функции ϕ найдётся вектор ν ∈ Rn такой, что ϕ(x ) = hνν, x i. Заметим, что вектор ν не параллелен ни одному ребру многогранника P n . Теперь мы будем рассматривать ϕ как функцию высоты на P n . При помощи ϕ мы превратим 1-остов многогранника P n в ориентированный граф, направляя каждое ребро так, чтобы функция ϕ возрастала вдоль него. yXX X 3] XXX J  J XXX   ind = 3 J XXX  J  BM  J B  J B  J B I
@ @
J B @
JX B yXX @
XXX 6  B XXXB @ ind = 2
@
 @ K A  A @  A @  ind = 1 A BM  B A ν  B A Y H  HH B  H B HH  B H HH B :  H B ind = 0 HHB Рис. 2: Ориентированный 1-остов многогранника P и индексы вершин. Для каждой вершины v многогранника P n определим ее индекс ind(v) как число рёбер, входящих в v. Обозначим число вершин индекса i через Iν (i). Мы утверждаем, что Iν (i) = hn−i . Действительно, каждая грань P n имеет единственную верхнюю вершину (максимум функции высоты ϕ, ограниченной на грань) и единственную ниж-

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нюю вершину (минимум ϕ). Пусть F k — некоторая k-грань и vF — её верхняя вершина. Так как многогранник P n простой, в вершине vF сходятся в точности k рёбер из грани F k , т.е. ind(vF ) > k. С другой стороны,
каждая вершина индекса q > k является верхней вершиной для kq граней размерности k. Отсюда следует, что fk может быть вычислено как X q  fk = Iν (q). k q>k

Тождество (7) показывает, что Iν (q) = hn−q , что и требовалось. В частности, число Iν (q) не зависит от ν . С другой стороны, так как indν (v) = n − ind−ν (v) для любой вершины v, мы имеем hn−q = Iν (q) = I−ν (n − q) = hq .

Предложение 8 (Кли [Kl]). Соотношения Дена–Соммервилля являются наиболее общими линейными уравнениями, которым удовлетворяют f -векторы всех простых (или симплициальных) многогранников. Доказательство. В [Kl] это утверждение было доказано непосредственно, в терминах f -векторов. Однако, использование h-векторов существенно упрощает доказательство. Достаточно показать, что аффинная оболочка h-векторов   (h0 , h1 , . . . , hn ) простых n-многогранников представляет собой n2 -мерную плоскость (напомним, что h0 = 1   всегда). Это можно сделать, например, указав n2 +1 простых многогранников с аффинно независимыми h-векторами. Положим Qk :=   ∆k ×∆n−k , k = 0, 1 . . . , n2 . Так как h(∆k ; t) = 1+t+. . .+tk , получаем h(Qk ; t) =

1 − tk+1 1 − tn−k+1 · . 1−t 1−t

Отcюда вытекает, что разность h(Qk+1 ; t)−h(Qk ; t) представляется n в виде суммы tk+1 и членов более высокого порядка, k = 0, 1, . . . , 2 −   1. Таким образом, векторы h(Qk ), k = 0, 1, . . . , n2 , аффинно независимы.

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Пример. Каждая вершина простого n-многогранника P n принадлежит в точности n рёбрам и каждое ребро соединяет две вершины. Отсюда вытекает следующее линейное соотношение на компоненты f -вектора: 2f1 = nf0 . (8) В силу предложения 8, это равенство должно вытекать из соотношений Дена–Соммервилля. Из (8) и формулы Эйлера следует, что f -вектор простого (или симплициального) 3-многогранника полностью определяется числом гиперграней (или вершин), а именно, f (P 3 ) = (2f2 − 4, 3f2 − 6, f2 ).

(9)

Заметим также, что формула Эйлера является единственным линейным соотношением, которому удовлетворяют векторы граней произвольных выпуклых многогранников. (Доказательство аналогично приведённому в предложении 8 — предъявляется достаточное количество многогранников с аффинно независимыми векторами граней.) Условия, полностью характеризующие f -векторы простых (или симплициальных) многогранников, ныне известные как g-теорема, были впервые сформулированы в виде гипотезы П. Макмюлленом [McM1] в 1970 и доказаны Р. Стенли [St2] (необходимость) и Биллерой и Ли [BL] (достаточность) в 1980. Кроме соотношений Дена– Соммервилля, g-теорема содержит две системы неравенств, одна из которых линейна, а другая — нелинейна. Для того, чтобы полностью сформулировать g-теорему нам понадобится следующая конструкция. Определение 9. Для любой пары натуральных чисел a, i существует единственное биномиальное i-разложение числа a вида a=

ai i




+

ai−1 i−1




+ ... +

aj j




,

где ai > ai−1 > . . . > aj > j > 1. Определим ahii =

ai +1 i+1




+

ai−1 +1 i




+ ... +

aj +1 j+1




,

0hii = 0.

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Пример 1. 1. При a > 0, ah1i = a+1 2 . 2. Если i > a, то биномиальное разложение имеет вид


i−a+1 a = ii + i−1 i−1 + . . . + i−a+1 = 1 + . . . + 1, и, следовательно, ahii = a. 3. Пусть a = 28, i = 4. Соответствующие биномиальное разложение есть


28 = 64 + 53 + 32 . Следовательно, 28h4i =

7 5




+

6 4




+

4 3




= 40.

Теорема 10 (g-теорема). Целочисленный вектор (f0 , f1 , . . . , fn−1 ) является f -вектором некоторого простого n-многогранника тогда и только тогда, когда соответствующая последовательность (h0 , . . . , hn ), определяемая (5), удовлетворяет следующим трём условиям: а) hi = hn−i , i = 0, . . . , n (соотношения Дена–Соммервилля);   б) h0 6 h1 6 . . . 6 hn , i = 0, 1, . . . , n2 ; 2

в) h0 = 1, hi+1 − hi 6 (hi − hi−1 )hii , i = 1, . . . ,

n 2

− 1.

Замечание. Очевидно, те же самые условия характеризуют и f векторы симплициальных многогранников. Определение 11. g-вектор многогранника P определяется следующим образом: n g0 = 1, gi = hi − hi−1 , 1 ≤ i ≤ [ ] 2 В терминах g-векторов последняя теорема формулируется проще. Теорема. Вектор (1, g1 , . . . , g[n/2] ) является g-вектором простого многогранника ⇔ 1. gi ≥ 0 hii

2. gi+1 ≤ gi

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Пример. 1. Первое неравенство h0 6 h1 части б) в g-теореме эквавалентно следующему: fn−1 = m > n + 1. Это лишь выражает тот факт, что для того, чтобы ограничить многогранник в Rn требуется как минимум n + 1 гиперплоскостей. 2. Принимая во внимание соотношение
h2 = n2 − (n − 1)fn−1 + fn−2 (см. (6)), мы можем переписать первое неравенство h2 − h1 6 (h1 − h0 )h1i части в) в g-теореме как

n+1 − nfn−1 + fn−2 6 fn−12 −n . 2 (см. пример 1.1). Это эквивалентно оценке сверху
fn−2 6 fn−1 , 2 которая выражает тот факт, что любые две гиперграни пересекаются не более чем по одной грани коразмерности два. В терминах двойственного симплициального многогранника, две вершины могут быть соединены не более чем одним ребром. 3. Второе неравенство h1 6 h2 (при n > 4) части б) в g-теореме эквивалентно следующему:
fn−2 > nfn−1 − n+1 2 . Это есть первое (и наиболее существенное) из неравенств в знаменитой Гипотезе о Нижней Границе для простых многогранников (см. теорему 17 ниже). Таким образом, первые две координаты h-векторов простых многогранников P n , n > 4, всегда попадают в область между двумя кривыми h2 = h1 (h21 +1) и h2 = h1 на (h1 , h2 )-плоскости (см. рис. 3). Заметим, что наиболее общими линейными неравенствами, которым удовлетворяют точки в этой области являются h1 > 1 и h2 > h1 . Задача характеризации f -векторов f (P n ) = (f0 , f1 , . . . , fn−1 ) произвольных выпуклых многогранников имеет богатую историю (здесь мы считаем, что fi есть число i-мерных граней в P n ). В общем случае не удаётся получить столь полных результатов, как для симплициальных и простых многогранников. В 1906 г. Штейницем было

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h2 =

27

h1 (h1 +1) 2

h2 = h1

h1

0 Рис. 3: (h1 , h2 )-область, n > 4.

получено полное описание множества f -векторов f (P 3 ) = (f0 , f1 , f2 ) трёхмерных выпуклых многогранников. Это множество представляет собой множество целых точек в двумерном выпуклом многогранном конусе, определяемом следующими тремя условиями: f0 − f1 + f2 = 2,

f2 − 4 6 2(f0 − 4),

f0 − 4 6 2(f2 − 4).

Первое неравенство обращается в равенство тогда и только тогда, когда P 3 — симплициальный многогранник (см. (9)), а второе — когда P 3 является простым многогранником. Уже в размерности 4 проблема характеризации f -векторов многогранников далека от завершения. Анализ положения дел в этой области за прошедшие 100 лет дан в докладе Циглера на международном конгрессе математиков в Пекине 2002 г.

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Теоремы о Верхней и Нижней Границе

Следующее утверждение, ныне известное как Теорема о Верхней Границе (ТВГ), было предложено Моцкиным в 1957 и доказано П. Мак-

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мюлленом [McM0] в 1970. Теорема 12 (ТВГ для симплициальных многогранников). Среди всех симплициальных n-многогранников S с m вершинам, циклический многогранник C n (m) (пример 2) имеет максимальное число i-граней, 2 6 i 6 n − 1. Таким образом, если f0 (S) = m, то
fi (S) 6 fi C n (m) для i = 2, . . . , n − 1. Равенство достигается тогда и только тогда, когда S является смежностным многогранником (см. определение 5). Заметим, что так как C n (m) — смежностный многогранник,

  m fi C n (m) = i+1 для i = 0, . . . , n2 − 1. В силу соотношений Дена–Соммервилля, это целиком определяет f вектор многогранника C n (m). Явные формулы для остальных компонент вектора f (C n (m)) даются следующей леммой. Лемма 13. Число i-граней циклического многогранника C n (m) (или произвольного смежностного n-многогранника с m вершинами) есть


fi C n (m) =

n 2 X

q n−1−i




q=0

где мы предполагаем

m−n+q−1 q




+

n−1 2 X

n−p i+1−p




m−n+p−1 p




,

p=0 p q




= 0 при p < q или q < 0.

Следствие 14. ТВГ для симплициальных многогранников (см. теорему 12) эквивалентна следующим неравенствам для h-вектора симплициального многогранника S с m вершинами:
  hi (S) 6 hi (C n (m)) = m−n+i−1 , i = 0, . . . , n2 . i Последнее утверждение было одним из ключевых наблюдений в оригинальном доказательстве ТВГ для симплициальных многогранников, данном Макмюлленом. ТВГ верна для произвольных выпуклых многогранников, не обязательно симплициальных. А именно, циклический многогранник

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C n (m) имеет максимальное возможное число i-граней среди всех выпуклых n-многогранников с m вершинами. Идея доказательства этого утверждения (принадлежащего Кли и Макмюллену, см. [Zi, лемма 8.24]) заключается в следующем. Немного "пошевелив"вершины выпуклого многогранника, можно добиться того, что получится симплициальных многогранник с тем же числом вершин, который в каждой размерности имеет не меньше граней, чем исходный. Другим фундаментальным фактом теории выпуклых многогранников является Теорема о Нижней Границе (ТНГ) для симплициальных многогранников. Определение 15. Симплициальный n-многогранник S называется многогранником пирамидальной надстройки, если имеется последовательность n-многогранников S0 , S1 , . . . , Sk = S такая, что S0 есть n-симплекс, а Si+1 получается из Si добавлением пирамиды над некоторой его гипергранью. В комбинаторных терминах, многогранники пирамидальной надстройки получаются из симплекса применением нескольких последовательных звёздных подразделений гиперграней. Замечание 16. Добавление пирамиды (или звёздное подразделение гиперграни) является двойственной операцией к “усечению вершины” простого многогранника. Соответственно, многогранники, двойственные к многогранникам пирамидальной надстройки, называются многогранниками усечения. Теорема 17 (ТНГ для симплициальных многогранников). Для любого симплициального n-многогранника S (n > 3) c m = f0 вершинами имеют место неравенства

fi (S) > ni f0 − n+1 для i = 1, . . . , n − 2; i+1 i fn−1 (S) > (n − 1)f0 − (n + 1)(n − 2). Равенство достигается тогда и только тогда, когда S есть многогранник пирамидальной надстройки. Тот факт, что равенство в ТНГ достигается только для многогранников пирамидальной надстройки, был доказан в на основе gтеоремы. В отличие от ТВГ, возможности обобщения ТНГ на несимплициальные выпуклые многогранники очень мало изучены.

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В силу двойственности, ТВГ и ТНГ дают соответственно верхнюю и нижнюю оценки на число граней простого многогранника с данным числом гиперграней. Обе теоремы были доказаны приблизительно в одно и то же время (в 1970) и послужили основанием П. Макмюллену сформулировать g-теорему в виде гипотезы [McM1]. C другой стороны, как ТВГ, так и ТНГ являются следствиями gтеоремы. На самом деле, ТНГ вытекает из утверждений а) и б) теоремы 10, в то время как ТВГ следует из а) и в). Неравенства части б) g-теоремы, т.е. h0 6 h1 6 . . . 6 hn ,

(10)

2

были предложены в [McMW] как обобщение ТНГ для симплициальных многогранников. Вопрос о справедливости этих неравенств для целого ряда более общих комбинаторных объектов в настоящее время известен под названием “Обобщённая гипотеза о нижней границе” (ОГНГ).

5

Одномерный остов многогранников

Теорема 18 (G. Kalai). Для простого многогранника P граф Γ(P ) однозначно восстанавливает комбинаторную структуру P . Иначе говоря, комбинаторная эквивалентность простых многогранников равносильны эквивалентности их графов. Заметим, что Γ(P ) - n-регулярен, т.е. любая его вершина имеет валентность n. Рассмотрим ориентации Γ(P ) без циклов. Ориентацию Γ(P ) называем хорошей, если любая грань P имеет ровно 1 сток (точку, куда ребра только входят). Лемма 19. Набор вершин {vi }I порождает k-грань P ⇔ 1. Γ({vi }I ) - k-регулярный граф; 2. {vi }I - начальное множество для некоторой хорошей ориентации (стрелки из которого только выходят).

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Доказательство. ⇒ 1) - следствие простоты. 2) - если набор порождает грань F , то поместив ее "вниз получим хорошую ориентацию, направив стрелки "вверх". ⇐ В множестве {vi }I существует хотя бы 1 сток X. Пусть F грань K, натянутая на k стрелок, приходящих в X. Докажем, что F = conv({vi }). Легко показать (пользуясь единственностью стока), что F ⊂ {vi }I . Получили k-регулярный подграф (граф грани F ) другого k-регулярного связного графа. Они обязаны совпадать. Лемма доказана. Осталась проблема: пока невозможно определить, является ли данная ориентация хорошей, т.к. не известно, какие вершины образуют грань. Пусть есть ориентация без циклов. Вычислим число пар #(грань F, сток грани F ). С одной стороны число граней K = f ≤ #(грань F, сток грани F ), причем равенство достигается в точности на хороших ориентациях. С другой стороны, пусть вершина X имеет индекс k , т.е. в нее входит k стрелок. На эти k ребер можно 2k способами натягивать грани, содержащие X в качестве стока. Поэтому X #(грань , сток) = 2k hOr k , k

где hOr k - число Pвершин индекса k относительно ориентации Or. Получили k 2k hOr k ≥ f (P ), и равенство достигается только на хороших ориентациях. Поэтому для определения хороших ориентаций, нужно перебрать все возможные ориентации без циклов, вычисP лить k 2k hOr k и найти минимальное значение. Ориентации с минимальным значением - и есть хорошие. • Не существует эффективного алгоритма восстановления многогранника по графу. • Для произвольных выпуклых многогранников последняя теорема неверна. Следующий результат имеет место только для размерности 3.

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Теорема 20 (Штейница). Граф Γ является 1-остовом выпуклого 3-многогранника ⇔ 1. Γ - простой, т.е. не имеет петель и кратных ребер. 2. Γ - 3-связный, т.е. остается связным при удалении любых 2х вершин. 3. Γ - планарный граф, т.е. реализуется на плоскости. Следствие 21. Граф Γ является 1-остовом простого 3-многогранника ⇔ 1. Γ - простой, т.е. не имеет петель и кратных ребер. 2. Γ - 3-связный, т.е. остается связным при удалении любых 2х вершин. 3. Γ - планарный граф, т.е. реализуется на плоскости. 4. Γ - 3-регулярный граф, т.е. все его вершины валентности 3. Уравнения f0 = 2f2 − 4 и f2 = 2f0 − 4 эквивалентны соотнощениям Дена-Соммервилля для простых и симплициальных многогранников. Обозначим через P2n абелеву группу, порожденную комбинаторными простыми Pn-мерными многогранниками. P2n обозначим градуированное коммутативное асЧерез P = n>0

социативное кольцо, в котором умножение соответствует прямому произведению простых многогранников, а единица P 0 = (точка). Обозначим через m = m(P n ) число гиперграней многогранника n P и положим ν(P n ) = m − n. Определен линейный оператор d : P → P, сопоставляющий простому многограннику P n сумму всех его гиперграней, т.е. d : P2n → P2(n−1) . Оператор d является дифференцированием кольца P, так как d(P1n1 × P2n2 ) = (dP1n1 )P2n2 + P1n1 (dP2n2 ). Например, d∆n = (n + 1)∆n−1 , ν(∆n ) = 1, т.е. ν(d∆n ) = ν(∆n ), dI n = n(dI)I n−1 = 2nI n−1 , ν(I n ) = n, ν(dI n ) = ν(I n ) − 1.

(11)

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Введем F - и H-полиномы двух переменных: F (P )(α, t) = αn + fn−1 αn−1 t + · · · + f1 αtn−1 + f0 tn , H(P )(α, t) = F (P )(α − t, t) = h0 αn + h1 αn−1 t + · · · + hn−1 αtn−1 + hn tn . Классические f - и h- полиномы получаются из данных подстановкой единицы вместо одного из параметров: f (P )(t) = F (P )(t, 1). h(P )(t) = H(P )(t, 1); Теорема 22. Линейное отображение F : P −→ Z[α, t] : P n −→ F (P n ) является кольцевым гомоморфизмом таким, что F (dP n ) =

∂ F (P n ). ∂t

(12)

Доказательство. Мультипликативность отображения F вытекает из того факта, что любая грань размерности s многогранника P1n1 ×P2n2 является произведением некоторой грани (скажем, размерности i) многогранника P1 на грань размерности j многогранника P2 , где i + j = s. Доказательство формулы (12) использует тот факт, что каждая k-мерная грань простого многоранника является пересечением точно n − k граней размерности n − 1. Следствие 23. F (I n ) = (α + 2t)n , F (∆n ) =

(α + t)n+1 − tn+1 . α

(13) (14)

Доказательство. Имеем F (I) = α + 2t. Следовательно, F (I n ) = F (I)n = (α + 2t)n . Формулу (14) докажем индукцией по n. Для n = 1 формула (14) верна. Допустим, что она верна для всех k 6 n. Тогда
∂ n+2 F (∆n+1 ) = F (d∆n+1 ) = (n + 2)F (∆n ) = (α + t)n+1 − tn+1 . ∂t α

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Следовательно, F (∆n+1 ) =


1 (α + t)n+2 − tn+2 + c(α). α

Используя, что F (∆n+1 (α, 0)) = αn+1 , получаем c(α) = 0. Отметим, что перечисляющий полином F (P n ) можно ввести для любого выпуклого многогранника P n ⊂ RN , но если P n не является простым, то формула (12) не верна. Вывод формулы (14) служит первым примером использования формулы (12). В качестве следующего приложения этой формулы получим теорему единственности для введенного выше преобразования f . Теорема 24. Пусть F : P −→ Z[α, t] : P n −→ F (P n )(α, t) — линейное отображение такое, что F (dP n ) =

∂ F (P n ), ∂t

F (P n ) t=0 = αn .

Тогда F (P n ) = f (P n ). Доказательство. Имеем F (P 0 ) = 1 = F (P 0 ). Допустим, что утверждение верно для всех k 6 n. Тогда F (dP n+1 )(α, t) = F (dP n+1 )(α, t) для всех простых (n + 1)-мерных многогранников. Получаем ∂ ∂ F (P n+1 ) = F (P n+1 ). ∂t ∂t Следовательно, F (P n+1 )(α, t) = F (P n+1 )(α, t) + c(α). Полагая t = 0, получаем αn+1 = αn+1 + c(α), т.е. c(α) = 0.

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Соотношения Дена-Соммервиля

Теорема 25. Для любого простого многогранника P n полином F (P n ) инвариантен относительно замены переменных (α, t) → (−α, α+t), т.е. F (P n )(α, t) = F (P n )(−α, α + t). (15) Доказательство. Имеем F (P 0 )(α, t) = 1 = F (P 0 )(−α, α + t). Допустим, что тождество (15) выполняется для всех k 6 n. Так как dP n+1 является суммой простых n-мерных многогранников, то F (dP n+1 )(α, t) = F (dP n+1 )(−α, α + t). Получаем ∂ ∂ F (P n+1 )(α, t) = F (P n+1 )(−α, α + t). ∂t ∂t Следовательно, F (P n+1 )(α, t) − F (P n+1 )(−α, α + t) = c(α). Полагая t = 0, получаем h  i αn+1 1 − (−1)n+1 + (−1)n fn + · · · + f0 = c(α). Граница ∂P n+1 простого многогранника представляет собой n-мерную сферу S n , разбитую на клетки — грани многогранника P n+1 . Следовательно, c(α) = 0, так как согласно классической формуле Эйлера– Пуанкаре для сферы S n мы имеем f0 − f1 + · · · + (−1)n fn = 1 + (−1)n .

Таким образом, мы получили новое доказательство классических соотношений Дена–Соммервиля.

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H-полиномы

Теорема 26. Для любого простого многогранника P n 1) H-полином H(P n )(α, t) является симметрической функцией переменных α и t, т.е. H(P n )(α, t) = H(P n )(t, α); 2) имеет место соотношение H(dP n )(α, t) = ∂H(P n )(α, t), где ∂ =

∂ ∂α

+

∂ ∂t

(16)

.

Так как переход от F -полинома к H-полиному задается заменой переменных (α, t) → (α − t, t), то мы получаем кольцевой гомоморфизм H : P −→ Z[α, t] : P n −→ H(P n )(α, t). Теорема 27. 1. Образ гомоморфизма H лежит в подкольце, порожденном полиномами H(∆1 ) = α + t

и

H(∆2 ) = α2 + αt + t2 .

2. Пусть ˜ : P −→ Z[α, t] H — линейное отображение, такое что n ˜ ˜ n ), H(dP ) = ∂ H(P

˜ n ) H(P = αn , t=0

∂ ∂ ˜ = H, т.е. H(P ˜ n ) = H(P n ) для любого где ∂ = ∂α + ∂t . Тогда H простого многогранника P n .

Доказательство. Утверждение 1 следует из того, что H(P n )(α, t) = H(P n )(t, α). ˆ 0 ) = H(P 0 ). Утверждение 2 докажем по индукции. Имеем H(P n n ˜ Допустим, что H(P ) = H(P ) для любого n-мерного простого многогранника. Тогда n+1 ˜ n+1 ) = H(dP ˜ ∂ H(P ) = H(dP n+1 ) = ∂H(P n+1 ).

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Следовательно, ˜ n+1 ) = H(P n+1 ) + ϕ(α, t), H(P

(17)

где ϕ(α, t) полином, удовлетворяющий уравнению ∂ ∂ ϕ(α, t) + ϕ(α, t) = 0. ∂α ∂t Таким образом, ϕ(α, t) = c(α − t). Полагая в (17) t = 0, получаем αn+1 = αn+1 + c(α), т.е. c(α) = 0, и поэтому ϕ(α, t) = 0.

8

Ассоциэдр. Циклоэдр. Пермутоэдр.

Рассмотрим многоугольник с n + 3 вершинами Pn+3 . Частично упорядоченное множество граней ассоциэдра (многогранника Сташефа) изоморфно частично упорядоченному множеству триангуляцияй многоугольника. • Вершинам соответствуют триангуляции Pn+3 . • Вершины соединены ребром если триангуляции связаны флипом. • Гиперграни соответствуют диагоналям Pn+3 . • Гиперграни пересекаются, если соответствующие диагонали не пересекаются. Рассмотрим многоугольник с 2n + 2 вершинами P2n+2 . Частично упорядоченное множество граней циклоэдра (многогранника БоттаТаубса) изоморфно частично упорядоченному множеству центральносимметричных триангуляцияй многоугольника. • Вершинам соответствуют центрально-симметричные триангуляции P2n+2 . • Вершины соединены ребром если триангуляции связаны либо флипом диаметра, либо двумя симметричным флипами.

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Рис. 4: Двумерный ассоциэдр As2

Рис. 5: Трехмерный ассоциэдр As3

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Рис. 6: Двумерный циклоэдр Cy 2

Рис. 7: Трехмерный циклоэдр Cy 3 • Гиперграни соответствуют диаметрам и парам симметричных диагоналей P2n+2 . • Гиперграни пересекаются, если соответствующие диагонали не пересекаются. Пермутоэдр - многогранник, полученный из симплекса следующим образом. Сначала у симплекса срезаются вершины, далее у получившегося многогранника срезаются ребра исходного сиплекса, и т.д. по возрастанию размерности срезаемых граней. Таким образом, гиперграни пермутоэдра соответствуют граням симплекса. Гипер-

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Рис. 8: Трехмерный пермутоэдр P e3

грани пермутоэдра пересекаются если

• Гиперграни пермутоэдра соответствуют граням симплекса.

• Гиперграни пермутоэдра пересекаются, если соответствующие грани симплекса образуют цепь.

Используя данные определения, можно показать, следующее. dAsn =

X

(i + 2)Asi · Asj ;

(18)

i+j=n−1

X

dCy n = (n + 1)

Asi · Cy j ;

(19)

i+j=n−1 n

dP e =

X i+j=n−1



 n+1 P ei · P ej ; i+1

(20) (21)

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Рассмотрим специально подобранные производящие ряды обсуждаемых серий простых многогранников. Положим: ∆(x) =

X n>0

As(x) =

X

∆n

xn+1 ; (n + 1)!

I(x) =

Pe(x) =

n>0

In

n>0

Asn xn+2 ;

Cy(x) =

n>0

X

X

X

xn ; n!

Cy n

n>0

xn+1 ; n+1

n+1

Pen

x ; (n + 1)!

Продолжим оператор d : P → P по линейности до оператора d : P ⊗ Q[[x]] −→ P ⊗ Q[[x]]. Тогда из приведенных выше формул (16)–(19) для оператора d получаем теорему. Теорема 28. Имеют место следующие соотношения: d∆(x) = x∆(x) ; d dAs(x) = As(x) As(x) ; dx dPe(x) = Pe2 (x) ; .

9

dI(x) = 2xI(x) ; d dCy(x) = As(x) Cy(x) ; dx

Вычисления F -полиномов

Рассмотрим преобразование F : P −→ Z[α, t]. Пусть U (x) =

P

λn Pn xn+q ∈ P ⊗ Q[[x]]. Положим

n>0

Uf (α, t; x) =

∞ X

λn F (Pn )xn+q .

n=0

Применяя преобразование f к соотношениям из теоремы 16, получаем, согласно теореме 1, следующий результат:

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Теорема 29. Имеют место следующие дифференциальные уравнения: 1 ∂ ∆f = x∆f , ∆f (α, 0; x) = (eαx − 1). 1. ∂t α ∂ 2. If = 2xIf , If (α, 0; x) = eαx . ∂t ∂ x2 ∂ 3. Asf = Asf Asf , Asf (α, 0; x) = . ∂t ∂x 1 − αx ∂ ∂ 1 4. Cyf = Asf Cyf , Cyf (α, 0; x) = − ln(1 − αx). ∂t ∂x α 1 αx ∂ 2 Pef = Pef (x), Pef (α, 0; x) = (e − 1). 5. ∂t α Следствие 30. ∆f (α, t; x) =

1 (α+t)x (e − etx ); α

If (α, t; x) = e(α+2t)x .

Следствие 31. Ряд Asf (α, t; x) такой, что Asf (α, 0; x) = однозначно задается как решение квадратного уравнения 
 x + tU x + (α + t)U = U.

x2 1−αx ,

(22)

Доказательство. Уравнение 3 из теоремы 29 представляет собой уравнение Э.Хопфа — знаменитое квазилинейное уравнение. Следствие 32.
 1  ln 1 − α x + tAsf (α, t; x) . α Доказательство. Непосредственная проверка показывает, что решение уравнения Cyf (α, t; x) = −

∂ ∂ U (t, x) = U0 (t, x) U (t, x), U (0, x) = ϕ(x), ∂t ∂x
имеет вид U (t, x) = ϕ x + tU0 (t, x) , если U0 (t, x) — решение уравнения Э.Хопфа. Следствие 33. Pef (α, t; x) =

eαx − 1 . α − t(eαx − 1)

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Флаговые многогранники 10

γ-полиномы и гипотеза Гала

Полином H(P n )(α, t) простого многогранника P n является симметрической функцией от α и t, поэтому его можно переписать в виде [n/2]

H(P n )(α, t) =

X

γi an−2i bi ,

(23)

i=0

где a = α + t и b = αt. Из условия h0 = h1 = 1 получаем, что γ0 = 1. [n/2] P Полином γ(P n )(τ ) = γi τ i называется γ-полиномом. i=0 2 2 , как и выше, ) = 1 + (m − 4)τ , где Pm Например, γ(I) = 1 и γ(Pm выпуклый m-угольник. Простой многогранник P n называется флаговым, если любое множество его гиперграней Fi1 , . . . , Fik имеет непустое пересечение, как только любая пара из этого множества имеет непустое пересечение. Гипотеза Гала (см. [G]). Пусть P n простой флаговый многогранник. Тогда γ-полином γ(P n )(τ ) имеет неотрицательные коэффициенты. 2 Например, если m-угольник Pm флаговый, то m > 4. В этом случае гипотеза Гала верна. При n = 2q полагая α = 1 и t = −1, получаем согласно формуле (23) 2q X (−1)i hi = (−1)q γq . i=0

Следовательно, известная гипотеза Черни–Дэвиса (см. [BP]) эквивалентна условию γq > 0, т.е. частному случаю гипотезы Гала. Этот факт отмечен уже в работе [G], в которой поставлена гипотеза Гала и представлены результаты в ее поддержку. g-вектор простого многогранника P n вычисляется через его γвектор по формуле (см. [Bu1]) gi = (n − 2i + 1)

i X j=0

  1 n − 2j γj , n−i−j+1 i−j

i = 0, . . . , [n/2]. (24)

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Следовательно для простых многогранников P1 и P2 выполнено: n γi (P1 ) ≤ γi (P2 ), i = 0, . . . , [ ] 2 ⇓ n gi (P1 ) ≤ gi (P2 ), i = 0, . . . , [ ] 2 ⇓ hi (P1 ) ≤ hi (P2 ), i = 0, . . . , n ⇓ fi (P1 ) ≤ fi (P2 ), i = 0, . . . , n − 1

11

Комбинаторные 2-усеченные кубы

Предложение 34. Пусть многогранник Q получен из простого nмногогранника P срезкой грани G размерности k, тогда 1) H(Q) = H(P ) + αtH(G)H(∆n−k−2 ); 2) γ(Q) = γ(P ) + τ γ(G)γ(∆n−k−2 ). Доказательство. Срезка заменяет грань G на грань G × ∆n−k−1 , тогда F (Q) = F (P ) + tF (G)F (∆n−k−1 ) − tn−k F (G). Откуда: H(Q) = H(P ) + tH(G)H(∆n−k−1 ) − tn−k H(G) = = H(P ) + tH(G)(

n−k−1 X

αi tn−i − tn−k−1 ) =

i=0

= H(P ) + αtH(G)(

n−k−2 X

αi tn−i ) = H(P ) + αtH(G)H(∆n−k−2 ) =

i=0 n

=

[2] X i=0

γi (P )(α + t)i (α + t)n−2i +

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 k  [2] X + αt  γi (G)(αt)i (α + t)k−2i  × i=0



] [ n−k−2 2

×

X

 γj (∆n−k−2 )(αt)j (α + t)n−k−2−2j  =

j=0

n

=

[2] X

γi (P )(αt)i (α + t)n−2i +

i=0 k

+

n−k−2

[2] [ 2 X X i=0

]

γi (G)γj (∆n−k−2 )(αt)i+j+1 (α + t)d−2(i+j+1)

j=0

Предложение доказано. Определение 35. Простой многогранник P n называется 2-усеченным кубом, если он может быть получен из куба I n последовательностью срезок граней коразмерности 2 (граница двойственного многогранника получается из кросс-политопа последовательностью звездных подразделений вдоль ребер). Следствие 36. Пусть многогранник Q получен из простого многогранник P срезкой грани G коразмерности 2, тогда 1) H(Q) = H(P ) + αtH(G); 2) γ(Q) = γ(P ) + τ γ(G). Можно показать, что срезка грани коразмерности 2 переводит простой флаговый многогранник в простой флаговый многогранник. Следовательно, каждый 2-усеченный куб является флаговым многогранником. Заметим также, что все грани 2-усеченного куба также являются 2-усеченными кубами. Предложение 37. Гипотеза Гала верна для класса 2-усеченных кубов. Т.е. P , то γi (P ) ≥ 0. Доказательство. С помощью индукции по размерности многогранника P и индукции по числу граней, используя формулу γ(Q) = γ(P ) + τ γ(G), получаем доказательство.

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12

Нестоэдры и граф-ассоциэдры

Обозначим через [n] и [i, j] множества {1, . . . , n} и {i, . . . , j}.

12.1

Производящие множества

Определение 38. Система B непустых подмножеств [n + 1] называется производящим множеством на [n + 1], если выполнены следующие условия: 1) {i} ∈ B для всех i ∈ [n + 1]. 2) Пусть S1 , S2 ∈ B и S1 ∩ S2 6= ∅, тогда S1 ∪ S2 ∈ B; Производящее множество B называется связным, если [n + 1] ∈ B. Напомним, что граф называется простым, если он не содержит петель и кратных ребер. Определение 39. Пусть Γ - простой граф на множестве вершин [n + 1]. Графическим производящим множеством B(Γ) называется система всех непустых подмножеств S ⊆ [n + 1], для которых индуцированный подграф Γ|S на множестве вершин S является связным. Замечание 40. Производящее множество B(Γ) связно тогда и только тогда, когда связен граф Γ.

12.2

Нестоэдры

Пусть M1 и M2 - подмножества Rn . Суммой Минковского M1 и M2 называется следующее подмножество Rn : M1 + M2 = {x ∈ Rn : x = x1 + x2 , где x1 ∈ M1 , x2 ∈ M2 }. Если M1 и M2 являются выпуклыми многогранниками, то M1 + M2 также является выпуклым многогранником. Определение 41. Пусть ei , i = 1, . . . , n + 1 - конечные точки базисных векторов Rn+1 . Нестоэдр PB , соответствующий производящему множеству B, определяется следующим образом: X PB = ∆S , где ∆S = conv{ei , i ∈ S}. S∈B

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x2

47

6 1 s

s   )

M2 s

s

 M1 + M2 s

-

x1
Рис. 9: Сумма Минковского отрезков M1 = conv (0, 0), (1, 0) и M2 =
conv (0, 1), (1, 0) . 0

M1

1

2

Если B(Γ) - графическое производящее множество, то многогранник PΓ = PB(Γ) называется граф-ассоциэдром. Предложение 42. • Каждый нестоэдр является простым многогранником. • Каждый граф-ассоциэдр является флаговым многогранником. флаговым. Теорема 43. Каждый флаговый нестоэдр является 2-усеченным кубом, и, следовательно, удовлетворяет гипотезе Гала.

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( n + 1)

As n

( n + 1) K

Cy n

n+1

P en K

1,n

St n

Рис. 10: Примеры граф-ассоциэдров

{ 2}

+ { 1, 2}

+

+

{ 2, 3}

{ 1, 2, 3}

+ { 1}

+ { 1, 3}

=

P e2

+ { 3}

Рис. 11: Производящее множество B(K3 ) = 2[3] \∅ = {S ⊆ [3] : S 6= ∅}

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Список литературы [BL]

L. Billera and C. Lee, Sufficiency of McMullen’s conditions for f -vectors of simplicial polytopes, Bull. Amer. Math. Soc. 2 (1980), 181–185.

[BL]

L. Billera and C. Lee, Sufficiency of McMullen’s conditions for f -vectors of simplicial polytopes, Bull. Amer. Math. Soc. 2 (1980), 181–185.

[Bu1]

V. Buchstaber, Ring of simple polytopes and differential equations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, vol. 263, pp. 18-43, 2008.

[Bu2]

V. Buchstaber, Lectures on Toric Topology. Toric Topology Workshop, KAIST 2008, Trends in Mathematics, Information Center for Mathematical Sciences, vol. 11, no. 1, pp. 1-55, 2008.

[St2]

R. Stanley, The number of faces of simplicial convex polytope, Adv. Math. 35 (1980), no. 3, 236–238.

[BV1]

В. М. Бухштабер, В. Д. Володин, "Точные верхние и нижние границы для нестоэдров Изв. РАН. Сер. матем., 75:6 (2011), 17-46.

[BV2]

V.M.Buchstaber, V.D.Volodin Combinatorial 2-truncated cubes and applications, Associahedra, Tamari Lattices, and Related Structures, Tamari Memorial Festschrift, Progress in Mathematics, Vol. 299, pp 161186, 2012.

[BP]

В. М. Бухштабер, Т. Е. Панов, Торические действия в топологии и комбинаторике., М., МЦНМО, 2004, (272 стр).

[G]

S. Gal, Real root conjecture fails for five- and higher-dimensional spheres. Discrete & Computational Geometry, vol. 34, no. 2, pp. 269-284, 2005; arXiv:math/0501046v1.

[Kl]

V. Klee, A combinatorial analogue of Poincar´ e’s duality theorem, Canad. J. Math. 16 (1964), 517–531.

[McM0]

P. McMullen, The maximum numbers of faces of a convex polytope, Mathematika 17 (1970), 179–184.

[McM1]

P. McMullen, The numbers of faces of simplicial polytopes, Israel J. Math. 9 (1971), 559–570.

[McM2]

P. McMullen, On simple polytopes, Invent. Math. 113 (1993), no. 2, 419– 444.

[McM96]

P. McMullen, Weights on polytopes, Discrete Comput. Geom. 15 (1996), no. 4, 363–388.

[McMSh]

P. McMullen and G. Shephard, Convex Polytopes and the Upper Bound Conjecture, London Maht. Soc. Lecture Notes Series 3, Cambridge University Press, Cambridge, 1971.

[McMW]

P. McMullen and D. W. Walkup. A generalised lower bound conjecture for simplicial polytopes, Mathematika 18 (1971), 264–273.

[St0]

R. Stanley, The Upper Bound Conjecture and Cohen–Macaulary rings, Studies in Applied Math. 54 (1975), no. 2, 135–142.

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[St1]

R. Stanley, Hilbert functions of graded algebras, Adv. Math. 28 (1978), no. 1, 57–83.

[Zi]

G. Ziegler, Lectures on Polytopes. Springer-Verlag, 1995. (Graduate Texts in Math. V.152).

Математический институт им. В. А. Стеклова РАН, г. Москва Лаборатория дискретной и вычислительной геометрии им. Делоне, г. Ярославль E-mail adress: [email protected]

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On Kneser’s problems V. L. Dolnikov

1

Knezer’s conjecture
• Definition. Let [n] denote the system of all r-element subsets r of the set [n] = {1, 2, . . . , n}. The Kneser graph KG(n, r) has the

T 0 vertex set [n] and edge set {v, v 0 } : v, v 0 ∈ [n] v = ∅}. r r ,v • The first interesting example of Kneser graph is KG(5, 2), which is called Petersen graph and this graph serves as a (counter)example for many questions in Graph Theory.

Fig. 1: Petersen graph.

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Fig. 2: Martin Kneser. • Definition. We recall that a (proper)k-coloring of a graph G = (V, E) is a mapping s : V → [k] such that c(v) 6= c(v 0 ) whenever {v, v 0 } ∈ E is an edge. The chromatic number of G, denote by χ(G), is a smallest k such that G has a k-coloring. • Kneser conjectured that χ(KG(n, r)) ≥ n − 2r + 2, n ≥ 2r ≥ 2.

Fig. 3: Knezer’s conjecture. • It is not hard to see that χ(KG(n, r)) ≤ n − 2r + 2, n ≥ 2r ≥ 2.

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• Indeed, we color the vertex v of Kneser graph in the color 1 if 1 ∈ v, in the color 2 if 2 ∈
v,..., in the color n − 2r + 1 if n − 2r + 1 ∈ v. If a vertex v ∈ [n] is not colored, then v ⊂ {n − 2r + 2, . . . , n}. r Evidently, if v, v 0 ⊂ {n − 2r + 2, . . . , n}, then v ∩ v 0 6= ∅. Therefore all such vertices we may color in the color n − 2r + 2. • Case r = 2. The proof that χ(KG(n, 2)) ≥ n − 2, n ≥ 3, is very simple. • The proof that χ(KG(n, 2)) ≥ n − 2, n ≥ 3, by induction on n. For n = 3, there is nothing to prove. • Let the Kneser graph KG(n, 2), n ≥ 4, is colored in k colors. Let us consider two cases.
• Case 1.: There exists i, 1 ≤ i ≤ n, and v1 , v2 , v3 ∈ [n] such that 2 x ∈ v1 ∩ v2 ∩ v3 and colors c(v1 ), c(v2 ), c(v3 ) of v1 , v2 , v3 are the same. Clearly, if c(v) = c(v1 ), then x ∈ v.
Therefore all v ∈ [n]/{i} are colored in k − 1 colors. By the 2 induction assumption, k − 1 ≥ n − 3. That is k ≥ n − 2.
• Case 2.: Every i ∈ [n] contains at most two sets [n] in of identical 2
n(n−1) n . But if n ≥ 4, then it is colors. Then 3k ≥ 2 , i.e. k ≥ 6 not hard to prove that n(n−1) ≥ n − 2. 6 • M. R. Garey and D. S. Johnson have solved the problem for r = 3 (The Complexity of Near Optimal Graph Coloring, J. Assoc. Comp. Mach., 23 (1976), 43-49. 1976) such elementary methods.

2

Borsuk - Ulam theorem and solutions of Knezer’s conjecture

This was proved in 1978 by Lov´ asz, as one of the earliest and most spectacular applications of topological methods in combinatorics. Knezer – Lov´ asz Theorem. If n ≥ 2r − 1, r > 0, then χ(KG(n, r)) = n − 2r + 2.

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• Let v ∈

[n] r




. Consider a characteristic point of v in Rd cv = (c1 , c2 , . . . , cd ),

where ci = 1 if i ∈ v, and ci = 0 if i ∈ / v. √
• If v ∩ v 0 = ∅, v, v 0 ∈ [d] , then d(v, v 0 ) = 2r, and if v ∩ v 0 6= ∅, r √ √ then d(v, v 0 ) ≤ 2r − 2 < 2r, where d(x, y) is Euclidean distance between points x, y. • Therefore Kneser’s problem is Borsuk’s problem for the point set Kn,r = {cv }v∈([n]) . r

• The Borsuk’s number, denote by B(d), is a smallest k such that every set A ⊂ Rd of diameter ≤ 1 can divide into k parts A1 , . . . , Ak of diameters < 1. • K.Borsuk conjectured that B(d) ≤ d + 1. G. Kalai and J. Kahn have disproved Borsuk’s conjecture in 1994. Borsuk’S conjecture appeared from the following topological theorem. Borsuk - Ulam - Lyusternik - Shnirelman Theorem. For d > 0 the following statement are equivalent: • (i) For every continuous mapping f : S d → Rd (S d is d-dimensional unit sphere) there exists a point x ∈ S d such that f (x) = f (−x). • (ii) For every antipode-preserving map f : S d → Rd there is a point x ∈ S d satisfying f (x) = 0. • (iii) There is no antipode-preserving map f : S d → S d−1 . • (iv) Let A1 , . . . , Ad+1 be a covering of S d by closed(open) sets Ai . Then there exists i such that Ai ∩ (−Ai ) 6= ∅. This theorem is one of the most applied theorems in topology. It was conjectured by Ulam at the Scottish Caffe in Lvov. Applications range from combinatorics to differential equations and even economics. The theorem proven in one form by Borsuk in 1933 has many equivalent

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formulations. One of these was first proven by Lyusternik and Shnirelman in 1930. No wonder that Borsuk-Ulam theorem played a main part in solutions of Kneser conjecture. • The first proof was suggested by 1.L.Lovasz (Kneser’s conjecture, chromatic number and homotopy, J.Comb.Theory Ser.A 25(1978), 319 – 324) as one of the earliest and most spectacular applications of topological methods in combinatorics. • Several other proofs have been published since then, all of them topological; among them, at least those of • 2. I.Barany A short proof of Kneser’s conjecture, J. Combinatorial Theory Ser.A 25 (1978), 325 – 326. , • 3. V.L. Dol’nikov Transversals of families of sets, In Studies in the theory of functions of several real variables (Russian), Yaroslav. State Univ., Yaroslavl’, 1981. 30 – 36, • 4. K. S.Sarkaria: A generalized Kneser conjecture, J. Combinatorial Theory Ser.B 49 (1990), 236 – 240, can be regarded as substantially different from each other and from L.Lovasz original proof. The proof of I.Barany in [2] bases on Borsuk – Ulam theorem and on the following lemma. Gale’s Lemma (1954). For every d > 0 and every k . 1, there exists a set X ⊂ S d of 2k + d points such that every open hemisphere of S d contains at least k points of X. Schrijver, using this ideas of I.Barany, considered the following problem.
Definition (Schrijver graph). Let us call a subset S of [n] stable if r it does not contain any two adjacent elements modulo n (if i ∈ S, then i+1 ∈ / S, and if n ∈ S, then 1 ∈ / S). In other words, S corresponds to
an independent set in the cycle C n . We denote by [n] r stab the family of stable r-subsets of [n].

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The Schrijver graph is SGn,r

  [n] := KG r stab

. It is an induced subgraph of the Kneser graph KGn,r , and as it turns out, it has the same chromatic number. For example, for KG5,2 , the Petersen graph, SG5,2 is a 5-cycle. Schrijver Theorem (1977). For all n ≥ 2r − 1 ≥ 0, we have χ(SGn,r ) = χ(KGn,r ) = n − 2r + 2. In fact, Schrijver showed that SGn,r is a vertex-critical subgraph of KGn,r that is, the chromatic number decreases if any single vertex (stable r-set) from SGn,r is deleted. The proof of [3] (1981) bases on the following theorem. Transversal theorem (1981). Suppose F1 , F2 , . . . Fk are families of compact convex sets in Rd . If for any i, 1 ≤ i ≤ k, every d − k + 2 or fewer members of Pi have a common point, then there is always a (k − 1)-flat that intersects all sets from all families. For k = d this theorem are equivalent to Borsuk – Ulam theorem. Let X be a finite ground set and let F ⊂ 2X be a set system (a hypergraph). The Kneser graph of F , denoted by KG(F), has F as the vertex set, and two sets F1 , F2 ∈ F are adjacent iff F1 ∩ F2 = ∅. In symbols, KG(F) = (F, {{F1 , F2 } : F1 , F2 ∈ F, F1 ∩ F2 = ∅}). Definition. Let the m-colorability defect, denoted by cdm (F ), be the minimum size of a subset Y ⊂ X such that the system of the sets of F that contain no points of Y is m-colorable. Remark. It easy to prove that cd2 (KG(n, r)) = n − 2r + 2. The following result is a straight corollary of Transversal Theorem. Theorem (1981). For any finite set system (X, F ), we have χ(KG(F )) ≥ cd2 (F )

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. Then clearly, χ(KG(n, r)) ≥ n − 2r + 2. We see that all proofs use topological methods. Then a problem about a combinatorial proof of Knezer’s conjecture appeared. The combinatorial version of Brower’s theorem is well-known. It is famous Sperner’s lemma. But there exists a discrete version of of Borsuk – Ulam theorem. Let B d denote octahedral ball in Rd (the unit ball of the l1 -norm), let Od−1 denote its boundary (the octahedral sphere), and let K0 be the natural triangulation of B n induced by the coordinate hyperplanes (the d-dimensional simplices are in one-to-one correspondence with the orthants in Rd ). Call a triangulation K of B d a special triangulation if it refines K0 and is antipodally symmetric around the origin. This statement is a discrete version of of Borsuk – Ulam theorem. Tucker’s lemma. Let K be a special triangulation of B n , and suppose that each vertex v of K is assigned a label λ(v) ∈ {+1, −1, +2, −2, . . . , +n, −n} in such a way that for the vertices of K lying in On−1 the labeling satisfies λ(−v) = −λ(v). Then there exists a 1-simplex (edge) of K which is complementary, i.e. its two vertices are labeled by opposite numbers. J. Matousek and G. Ziegler, using Tucker’s lemma, obtained discrete proofs (different) of this theorem.

3

Knezer’s problem for hypergraphs

There exist several generalizations of Knezer’s problem for hypergraphs. First generalization proposed P.Erd¨os. Definition of P.Erd¨ os for Kneser r-hypergraphs KGr (F ). Let F be a set system. Generalizing the notion of Kneser graph, we define the Kneser r-hypergraph V: The vertex set is F , and the edges are all rtuples of disjoint sets. For the Kneser r-hypergraph KGr (F ), we color

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the sets in F , and we want that no r disjoint sets get the same color. Phrased differently, we want a coloring of the vertices. Definition. We recall that a proper m-coloring of a hypergraph H is a mapping c : V (H) → [m] such that no edge of H is monochromatic. In the same way, we define the m-colorability defect cdm (F ). Alon, Frankl, and Lov´ asz established P.Erd¨os generalization of Kneser – Erd¨ os conjecture for such hypergraphs: If n > (m − 1)(r − 1) + rk, then

  [n] χ(KGr ( )) > m. k

For this purpose, they defined a box complex of an r-uniform hypergraph. I.Kriz generalized the theorem about the m-colorability defect cdm (F ) for Knezer – Erd¨ os hypergraphs. For any set system F , χ(KGr (F )) ≥

1 cdr (F ). r−1

N. Alon, P. Frankl, L. Lovґasz: The chromatic number of Kneser hypergraphs, Transactions Amer. Math. Soc., 298, pp. 359-370, 1986. I. Kriz. Equivariant cohomology and lower bounds for chromatic numbers. Transactions Amer. Math. Soc., 333:567–577, 1992. Other definition of Kneser r-hypergraph was supposed in [3]. Let F be a set system. Kneser l-hypergraph KG0l (F ): the vertex set is F , and the edges are all l-tuples without a common element. Conjecture 1(1981). If l ≤ r + 1, then   [n] lr 0 χ(KGl ( ) = n−] [+2, l−1 r where ]x[ — smallest integer number n such that n ≥ x.

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Definition.Denote by γ(n, k, r) the smallest number of points in a set A ⊂ Rn such that all convex hulls of all r-element subsets of A do not intersect a k-dimensional flat. Conjecture 2(1981). γ(n, k, r) = k+]

n−k+1 r[. n−k

Conjectures 1, 2 were proved for r = 2, 3.

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Optimal Networks A. O. Ivanov and A. A. Tuzhilin

The aim of this mini-course is to give an introduction in Optimal Networks theory. Optimal networks appear as solutions of the following natural problem: How to connect a finite set of points in a metric space in an optimal way? We cover three most natural types of optimal connection: spanning trees (connection without additional road forks), shortest trees and locally shortest trees, and minimal fillings.

1

Introduction: Optimal Connection

This mini-course was given in the First Yaroslavl Summer School on Discrete and Computational Geometry in August 2012, organized by International Delaunay Laboratory “Discrete and Computational Geometry” of Demidov Yaroslavl State University. We are very thankful to the organizers for a possibility to give lectures their and to publish this notes, and also for their warm hospitality during the Summer School. The real course consisted of three 1 hour lectures, but the division of these notes into sections is independent on the lectures structure. The video of the lectures can be found in the site of the Laboratory (http://dcglab.uniyar.ac.ru). The main reference is our books [1] and [2], and the paper [3] for Section 5. Our subject is optimal connection problems, a very popular and important kind of geometrical optimization problems. We all seek what is better, so optimization problems attract specialists during centuries. Geometrical optimization problems related to investigation of critical points of geometrical functionals, such as length, volume, energy, etc. The main example for us is the length functional, and the corresponding optimization problem consists in finding of length minimal connections. 60

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Figure 1: The shortest curves connecting a pair of points in Manhattan plane (left), and locally shortest but not shortest curve in this plane (right).

1.1

Connecting Two Points

If we have to points A and B in the Euclidean plane R2 , then, as we know from the elementary school, the shortest curve joining A and B is unique and coincides with the straight segment AB, so optimal connection problem is trivial in this case. But if we change the way of distance measuring and consider, for example, so-called Manhattan plane, i.e. the plane R2 with fixed standard coordinates (x, y) and the distance function ρ1 (A, B) = |a1 − b1 | + |a2 − b2 |, where A = (a1 , a2 ) and B = (b1 , b2 ), then it is not difficult to verify that in this case there are infinitely many shortest curves connecting A and B. Namely, if 0 ≤
a1 < b1 and 0 ≤ a2 < b2 , then any monotonic curve γ(t) = x(t), y(t) , t ∈ [0, 1], γ(0) = A, γ(1) = B, where functions x(t) and y(t) are monotonic, are the shortest, see Figure 1, left. Another new effect that can be observed in this example is as follows. In the Euclidean plane a curve such that each its sufficiently small piece is a shortest curve joining its ends (socalled locally shortest curve) is a shortest curve itself. In the Manhattan plane it is not so. The length of a locally shortest curve having the form of the letter Π, see Figure 1, right, can be evidently decreased. Similar effects can be observed in the surface of standard sphere S 2 ⊂ R3 . Here the shortest curve joining a pair of points is the lesser arc of the great circle (the cross-section of the sphere by a plane passing through the origin). Two opposite points are connected by infinitely many shortest curves, and if points A and B are not opposite, then the corresponding great circle is unique and it is partitioned into two arcs,

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both of them are locally shortest, one is the (unique) shortest, but the other one is not. (Really speaking, the difference with the Manhattan plane consists in the fact that for the case of the sphere any directional derivative of the length of any locally shortest arc with respect to its deformation preserving the ends is equal to zero). Exercise 1.1 For a pair of points on the surface of the cube describe shortest and locally shortest curves. Find out an infinite family of locally shortest curves having pairwise distinct lengths.

1.2

Connecting Many Points: Possible Approaches

Let us consider general situation, when we are given with a finite set M = {A1 , . . . , An } of points in a metric space (X, ρ), and we want to connect them in some optimal way in the sense of the total length of the connection. We are working under assumption that we already know how to connect pairs of points in (X, ρ), therefore we need just to organize the set of shortest curves in appropriate way. There are several natural statements of the problem, and we list here the most popular ones. 1.2.1

No Additional Forks Case: Spanning Trees

We do not allow additional forks, that is, we can switch between the shortest segments at the points from M only. As a result, we obtain a particular case of Graph Theory problem about minimal spanning trees in a connected weighted graph. We recall only necessary concepts of Graph Theory, the detail can be found, for example in [4]. Recall that a (simple) graph can be considered as a pair G = (V, E), consisting of a finite set V = {v1 , . . . , vn } of vertices and a finite set E = {e1 , . . . , em } of edges, where each edge ei is a two-element subset of V . If e = {v, v 0 }, then we say that v and v 0 are neighboring, edge e joins or connects them, the edge e and each of the vertices v and v 0 are incident. The number of vertices neighboring to a vertex v is called the degree of v and is denoted by deg v. A graph H = (VH , EH ) is said to be a subgraph of a graph G = (VG , EG ), if VH ⊂ VG and EH ⊂ EG . The subgraph H is called spanning, if VH = VG . A path γ in a graph G is a sequence vi1 , ei1 , vi2 . . . , eik vik+1 of its vertices and edges such that each edge eis connects vertices vis and

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vis+1 . We also say that the path γ connects the vertices vi1 and vik+1 which are said to be ending vertices of the path. A path is said to be cyclic, if its ending vertices coincide with each other. A cyclic path with pairwise distinct edges is referred as a simple cycle. A graph without simple cycles is said to be acyclic. A graph is said to be connected, if any its two vertices can be connected by a path. An acyclic connected graph is called a tree. If we are given with a function ω : E → R on the edge set of a graph G, then the pair (G, ω) is referred as a weighted graph. For
any subgraph H =P(VH , EH ) of a weighted graph G = (VG , EH ), ω the value ω(H) = e∈EH ω(e) is called the weight of H. Similarly, for any Pk path γ = vi1 , ei1 , vi2 . . . , eik vik+1 the value ω(γ) = s=1 ω(eis ) is called the weight of γ.
For a weighted connected graph G = (VG , EG ), ω with positive weight function ω, a spanning connected subgraph of minimal possible weight is called minimal spanning tree. The positivity of ω implies that such subgraph is acyclic, i.e. it is a tree indeed. The weight of any minimal spanning tree for (G, ω) is denoted by mst(G, ω). Optimal connection problem without additional forks can be considered as minimal spanning tree problem for a special graph. Let M = {A1 , . . . , An } be a finite set of points in a metric space (X, ρ) as above. Consider the complete graph K(M ) with vertex set M and edge set consisting of all two-element subsets of M . In other words, any two vertices Ai and Aj are connected by an edge in K(M ). By Ai Aj we denote the corresponding edge. The number of edges in K(M ) is, evidently, n(n−1)/2. We define the positive weight function ωρ (Ai Aj ) = ρ(Ai , Aj ). Then any minimal spanning tree T in K(M ) can be considered as a set of shortest curves in (X, ρ) joining corresponding points and forming a network in X connecting M without additional forks in an optimal way, i.e. with the least possible length. Such a network is called a minimal spanning tree for M in (X, ρ). Its total weight ωρ (T ) is called length and is denoted by mstX (M ). In Section ?? we speak about minimal spanning trees in more details. 1.2.2

Shortest tree: Fermat–Steiner Problem

But already P. Fermat and C. F. Gauss understood that additional forks can be profitable, i.e. can give an opportunity to decrease the length of

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Figure 2: Minimal spanning tree (left), shortest tree (center), and minimal filling, connecting the vertex set of regular triangle in Euclidean plane.

optimal connection. For example, see Figure 2, if we consider the vertex set M = {A1 , A2 , A3 } of a regular triangle with side 1 in the Euclidean plane, then the corresponding graph K(M ) consists of three edges of the same weight 1 and each minimal spanning tree consists of two edges, so mstR2 (M ) = 2. But if we add the center T of the triangle and consider the network consisting of√three straight segments A1 T , A2 T , A3 T , then √ its length is equal to 3 32 23 = 3 < 2, so it is shorter than the minimal spanning tree. This reasoning leads to the following general definition. Let M = {A1 , . . . , An } be a finite set of points in a metric space (X, ρ) as above. Consider a larger finite set N , M ⊂ N ⊂ X, and a minimal spanning tree for N in X. Then this tree contains M as a subset of its vertex set N , but also may contain some other additional vertices-forks. Such additional vertices are referred as Steiner points. Further, we define a value smtX (M ) = inf N :M ⊂N ⊂X mstX (N ) and call it the length of shortest tree connecting M or of Steiner minimal tree for M . If this infimum attains at some set N , then each minimal spanning tree for this N is called a shortest tree or a Steiner minimal tree connecting M . Famous Steiner problem is the problem of finding a shortest tree for a given finite subset of a metric space. We will speak about Steiner problem in more details in Section 3. The shortest tree for the vertex set of a regular triangle in the Euclidean plane is depicted in Figure 2.

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Minimizing over Different Ambient Spaces: Minimal Fillings

Shortest trees give the least possible length of connecting network for a given finite set in a fixed ambient space. But sometimes it s possible to decrease the length of connection by choosing another ambient space. Let M = {A1 , . . . , An } be a finite set of points in a metric space (X, ρ) as above, and consider M as a finite metric space with the distance function ρM obtained as the restriction of the distance function ρ. Consider an isometric embedding ϕ : (M, ρM ) → (Y, ρY ) of this finite metric space (M, ρM ) into
a (compact) metric space (Y, ρY ) and consider the value smtY ϕ(M ) . It could be less than smtX (M ). For example, the vertex set of the regular triangle with side 1 can be embedded into Manhattan plane as the set (−1/2, 0), (0, 1/2), (1/2, 0) , see Figure 2. Than the unique additional vertex of the shortest tree is the origin and the length √ of the tree is 3/2 < 3. So, for a finite metric
space M = (M, ρM ), consider the value mf(M) = inf ϕ smtY ϕ(M ) which is referred as weight of minimal filling of the finite metric space M. Minimal fillings can be naturally defined in terms of weighted graphs and can be considered as a generalization of Gromov’s concept of minimal fillings for Riemannian manifolds. We speak about them in more details in Section 5.

2

Minimal Spanning Trees

In this section we discuss minimal spanning trees construction in more details. As we have already mentioned above, in this case the problem can be stated in terms of Graph Theory for an arbitrary connected weighted graph. But geometrical interpretation permits to speed up the algorithms of Graph Theory.

2.1

General Case: Graph Theory Approach

We start with the Graph Theory problem of finding a minimal spanning tree in a connected weighted graph. It is not difficult to verify that direct enumeration of all possible spanning subtrees of a connected graph leads to an exponential algorithm. To see that, recall well-known Kirchhoff theorem counting the number of spanning subtrees. If G = (V, E) is a connected graph with

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enumerated vertex set V = {v1 , . . . , vn }, then its Kirchhoff matrix is defined as (n × n)-matrix BG = (bij ) with elements  deg vi if i = j, if {vi , vj } ∈ E, bij =  −1 0 otherwise. Then the following result based on elementary Graph Theory and Binet– Cauchy formula for determinant calculation is valid, see proof, for example in [4]. Theorem 2.1 (Kirchhoff ) For a connected graph G with n ≥ 2 vertices, the number of spanning subtrees is equal to the algebraic complement of any element of the Kirchhoff matrix BG . Example. Let G = Kn be the complete graph with n vertices. Than its Kirchhoff matrix has the following form:   n−1 −1 −1 · · · −1  −1 n − 1 −1 · · · −1    BKn =  . .. .. .. .. ..   . . . . . −1 −1 −1 · · · n − 1 The algebraic complement of the element bnn is equal to n−1 −1 ··· −1 1 · · · 1 −1 −1 · · · n − 1 · · · −1 −1 = .. .. .. .. .. .. .. . . . . . . . −1 −1 · · · n − 1 −1 · · · n − 1 =

1 0 .. .

1 n .. .

··· ··· .. .

0

0

···

=

= nn−2 , n 1 0 .. .

where the first equality is obtained by change of the first row by the sum of all the rows, and the second equality is obtained by change of the ith row, i ≥ 2, by the sum of it with the first row.

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Corollary 2.2 The complete graph with n vertices contains nn−2 spanning trees. Remark. Notice that this result is equivalent to Cayley Theorem saying that the total number of trees with n enumerated vertices is equal to nn−2 . But it is a surprising fact, that there exist polynomial algorithms constructing minimal spanning trees. Several such algorithms were discovered in 1960s. We tell about Kruskal’s algorithm. Similar Prim’s algorithm can be found in [4].
So, we are given with a connected weighted graph G = (V, E), ω with positive weight function ω. At the initial step of Kruckal algorithm we construct the graph T0 = (V, ∅) and put E0 = E. If the graph Ti−1 and the non-empty set Ei−1 ⊂ E, i ≤ n − 1, have been already constructed, then we choose in Ei−1 an edge ei of least possible weight and construct a new graph Ti = Ti−1 ∪ ei and also a new set Ei = {e ∈ E | e 6∈ Ti and Ti ∪ e is acyclic}. Algorithm stops when the graph Tn−1 is constructed1 . Theorem 2.3 (Kruskal) Under the above notations, the graph Tn−1 can be constructed for any connected weighted graph G = (G, ω), and moreover, Tn−1 is a minimal spanning tree in G. Proof. The set Ei is non-empty for all 0 ≤ i ≤ n − 2, because the corresponding subgraphs Ti are not connected (the graph Ti has n vertices and i edges), therefore all the graphs T1 , . . . , Tn−1 can be constructed. Further, all these graphs are acyclic due to the construction, and Tn−1 has n vertices and n − 1 edges, so it is a tree. To finish the proof it remains to show that the spanning tree Tn−1 ⊂ G is minimal. Since the graph G has a finite number of spanning trees, a minimal spanning tree does exist. Let T be a minimal spanning tree. We show that it can be reconstructed to the tree Tn−1 without changing the total weight, so Tn−1 is also a minimal spanning tree. To do this, recall that the edges of the tree Tn−1 are enumerated in accordance with the work of the algorithm. Denote them by e1 , . . . , en−1 1 Here the operation of adding an edge e to a graph G = (V, E) can be formally defined as follows: G ∪ e = (V, E ∪ {e}). Similarly, G \ e = (V, E \ {e}).

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as above, and assume that ek is the first one that does not belong to T . The graph T ∪ ek contains a unique cycle c ⊃ ek . This cycle c also contains an edge e not belonging to Tn−1 (otherwise c ⊂ Tn−1 , a contradiction). Consider the graph T 0 = T ∪ ek \ e. It is evidently a spanning tree in G, and therefore its weight is not less than the weight of the minimal spanning tree T , hence ω(T 0 ) = ω(T ) + ω(ek ) − ω(e) ≥ ω(T ), and thus, ω(ek ) ≥ ω(e). On the other hand, all the edges e1 , . . . , ek−1 belongs to T by our assumption. Therefore, the graph Tk−1 ∪ e is a subgraph of T and is acyclic, in particular. Hence, e ∈ Ek−1 so as ek ∈ Ek−1 . But the algorithm has chosen ek , hence ω(ek ) ≤ ω(e). Thus, ω(ek ) = ω(e), and so ω(T 0 ) = ω(T ), and therefore T 0 is a minimal spanning tree in G. But now T 0 contains the edges e1 , . . . , ek from Tn−1 . Repeating this procedure we reconstruct T to Tn−1 in the class of minimal spanning trees. Theorem is proved. Remark. For a connected weighted graph with n vertices and m edges the complexity of the Kruskal’s algorithm can be naturally estimated as mn ∼ n3 . The estimation can be improved to m log m ∼ n2 log n. The fastest non-randomized comparison-based algorithm with known complexity belongs to Bernard Chazelle [5]. It turns out that if the weight function is geometrical, then the algorithms can be improved.

2.2

Euclidean Case: Geometrical Approach

Now assume that M is a finite subset of the Euclidean plane R2 . It turns out that a minimal spanning tree for M in R2 can be constructed faster than the one for an abstract complete graph with n = |M | vertices by means of some geometrical reasonings. To do that we need to construct so called Voronoi partition of the plane, corresponding to M , and the Delaunay graph on M . It turns out that any minimal spanning tree for M in R2 is a subgraph of the Delaunay graph, see Figure 3, and the number of edges in this graph is linear with respect to n, so the standard Kruskal’s algorithm applied to it gives the complexity n log n instead of n2 log n for the complete graph with n vertices.

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Figure 3: Voronoi diagram (left) and Delaunay graph together with a minimal spanning tree (right) for a point set in the plane. Let us pass to details. Let M = {A1 , . . . , An } ⊂ R2 be a finite subset of the plain. The Voronoi cell of the point Ai is defined as  VorM (Ai ) = x ∈ R2 | kx − Ai k ≤ kx − Aj k for all j . The Voronoi cell for Ai is a convex polygonal domain which is equal to the intersection of the closed half-planes restricted by the perpendicular bisectors of the segments Ai Aj , j 6= i. It is easy to verify, that the intersection of any two Voronoi cells has no interior points and that ∪i VorM (Ai ) = R2 . This partition of the plane is referred as Voronoi partition or Voronoi diagram. Two cells VorM (Ai ) and VorM (Aj ) are said to be adjacent, if there intersection contains a straight segment. The Delaunay graph D(M ) is defined as the dual planar graph to the Voronoi diagram. More precisely, the vertex set of D(M ) is M , and to vertices Ai and Aj are connected by an edge, if and only if their Voronoi cells VorM (Ai ) and VorM (Aj ) are adjacent. The edges of the Delaunay graph are the corresponding straight segments. It is easy to verify, that if the set M is generic in the sense that no three points lie at a common straight line and no four points lie at a common circle, then the Delaunay graph D(M ) is a triangulation, i.e. its bounded faces are triangles. In general case some bounded faces could be inscribed polygons. Anyway, the number of edges of the graph D(M ) does not exceed 3n. It remains to prove the following key Lemma.

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Lemma 2.4 Any minimal spanning tree for M ⊂ R2 is a subgraph of the Delaunay graph D(M ). Proof. Let e = Ai Aj be an edge of a minimal spanning tree T for M . We have to show that the Voronoi cells VorM (Ai ) and VorM (Aj ) are adjacent. The graph T \ e consists of two connected components, and this partition generates a partition of the set M into two subsets, say M1 and M2 . Assume that Ai ∈ M1 and Aj ∈ M2 . The minimality of the spanning tree T implies that kAi , Aj k is equal to the distance between the sets M1 and M2 , where kAi , Aj k stands for the distance between Ai and Ak . By u we denote the middle point of the straight segment Ai Aj , and let Ak be another point from M . Assume that Ak ∈ M2 . Due to the previous remark, kAi , Ak k ≥ kAi , Aj k, therefore ku, Ak k ≥ kAi , Ak k − kAi , uk ≥ kAi Aj k − kAi , uk = = kAi , Aj k/2 = ku, Ai k = ku, Aj k. On the other hand, if ku, Ak k = ku, Ai k, then we have equalities in both above inequalities. The first one means that u lies at the straight segment Ai Ak , hence Ak lies at the ray Ai u. The second equality implies kAi Ak k = kAi Aj k, and so Ak = Aj , a contradiction. Thus, ku, Ak k > ku, Ai k, that is u does not belong to the cell VorM (Ak ) for k 6∈ {i, j}. Thus, u ∈ VorM (Ai ) ∩ VorM (Aj ), Since the inequality proved is strict, the same arguments remain valid for points lying close to u on the perpendicular bisector to the segment Ai Aj . Therefore, the intersection of the Voronoi cells VorM (Ai ) and VorM (Aj ) contains a straight segment, that is the cells are adjacent. Lemma is proved. Remark. The previous arguments work in any dimension. But the trouble is that starting from the dimension 3 the number of edges in the Delaunay graph need not be linear on the number of its vertices. Exercise 2.5 Verify that the same arguments can be applied to minimal spanning trees for a finite subset of Rn . Exercise 2.6 Give an example of a finite subset M ⊂ R3 such that the Delaunay graph D(M ) coincides with the complete graph K(M ).

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Problem 2.7 In what metric spaces similar geometrical approach also works? It definitely works for planar polygons with intrinsic metric, see [6].

3

Steiner Trees and Locally Minimal Networks

In this section we speak about shortest trees and locally shortest networks in more details. Besides necessary definitions we discuss local structure theorems, Melzak algorithm constructing locally minimal trees in the plane, global results concerning locally minimal binary trees in the plane (so called twisting number theory) and the particular case, locally minimal binary trees with convex boundaries (language of triangular tilings). The details concerning twisting number and tiling realization theory can be found in [2] or [1], and also in [7].

3.1

Fermat Problem

The idea that additional forks can help to decrease the length of a connecting network had been already clear to P. Fermat and his students. It seems that Fermat was the first, who stated the following optimization problem: for given three points A1 , A2 , and A3 in the plane find a point X minimizing the P sum of distances from the points Ai , i.e. minimize the function F (X) = i kAi , Xk. For the case when all the angles of the triangle A1 A2 A3 are less than or equal to 120◦ the solution was found by E. Torricelli and later by R. Simpson. The construction of Torricelli is as follows, see Figure 4. On the sides of the triangle A1 A2 A3 construct equilateral triangles Ai Aj A0k , {i, j, k} = {1, 2, 3}, such that they intersect the initial triangle only by the common sides. Then, as Torricelli proved, the circumscribing circles of these three triangles intersect in a point referred as Torricelli point T of the triangle A1 A2 A3 . If all the angles Ai are less than or equal to 120◦ , then T lies in the triangle A1 A2 A3 and gives the unique solution to the Fermat problem.2 Later Simpson proved that the straight 2 An elementary proof can be obtained by rotation R of a copy of the triangle around its vertex, say A1 , by 60◦ and considering the polygonal line L joining A2 , X, image R(X) of X under the rotation, and R(A3 ). The length of L is equal to

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Figure 4: Torricelli–Simpson construction, the case Ai ≤ 120◦ (left), and the case A3 > 120◦ (right). segments Ai A0i also pass through the Torricelli point, and the lengths of all these three segments are equal to F (T ). If one of the angles, say A3 , is more than 120◦ , then the Torricelli point is located outside the triangle and can not be the solution to Fermat problem. In this case the solution is X = A3 . Remark. So we see, that shortest tree for a triangle in the plane consists of straight segments meeting at the vertices by angles more than or equal to 120◦ . It turns out, that this 120◦ -property remains valid in much more general situation.

3.2

Local Structure Theorem and Locally Minimal Networks

Let M = {A1 , . . . , An } be a finite subset of Euclidean space RN , and T is a Steiner tree connecting M . Recall that we defined shortest trees as abstract graphs with vertex set in the ambient metric space. In the case of RN it is natural to model edges of such graph as straight segments joining corresponding points in the space. The configuration obtained is referred as a geometrical realization of the corresponding graph. Below, F (X), and minimal value of F (X) corresponds to the location of the X such that L is a straight segment.

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speaking about shortest trees in RN we will usually mean their geometrical realizations. The local structure of a shortest tree (more exactly of a geometrical realization of the tree) can be easily described. Theorem 3.1 (Local Structure) Let Γ be a shortest tree connecting a finite subset M = {A1 , . . . , An } in RN . Then 1. all edges of Γ are straight segments; 2. any vertex v ∈ Γ of degree 1 belongs to M ; 3. any two neighboring edges of Γ meet in common vertex by angle more than or equal to 120◦ ; 4. if the degree of a vertex v is equal to 2 and v 6∈ M , then the edges meet at v by 180◦ angle. Corollary 3.2 Let Γ be a shortest tree connecting a finite subset M = {A1 , . . . , An } in RN . Then the degree of any its vertex is at most 3, and if the degree of a vertex v equals to 3, then the edges meet at v by angles equal to 120◦ . Example. Let M be the vertex set of regular tetrahedron ∆ in R3 . Then the network consisting of four straight segments joining the vertices of the tetrahedra with its center O is not a shortest network. Indeed, since deg O = 4, then the angles between the edges meeting at O are less than 120◦ . The set M is connected by three different (but isometrical) shortest networks, each of which has two additional vertices of degree 3, see Figure 5. Theorem 3.1 can be just “word-by-word” extended to the case of Riemannian manifolds (we only need to change straight segments by geodesic segments) [1] and even to the case of Alexandrov spaces with bounded curvature. The case of normed spaces turned out to be more complicated (some general results can be found in [2]). A connected graph Γ in RN (in a Riemannian manifold) whose vertex set contains a finite subset M ⊂ RN is called a locally minimal network connecting M or with the boundary ∂Γ = M , if it satisfies Conditions (1)–(4) from Theorem 3.1. In the case of complete Riemannian manifolds such graphs are minimal “in small,” i.e. the following result holds, see [1].

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Figure 5: Non-shortest tree (left) and one of the shortest trees (right) for the vertex set of a regular tetrahedron. Theorem 3.3 (Minimality “in small”) Let Γ be a locally minimal network connecting a finite subset M of a complete Riemannian manifold W . Then each point P ∈ Γ possesses a neighborhood U in W , such that the network Γ ∩ U is a shortest network with the boundary (∂Γ ∩ U ) ∪ (Γ ∩ ∂U ). Remark. In the case of normed spaces Theorem 3.3 is not valid even for two-point sets, see example in Figure 1.

3.3

Melzak Algorithm and Steiner Problem Complexity

Let us return back to the case of Euclidean plane. It turns out that in this case the Torricelli–Simpson construction can be generalized to a geometrical algorithm, that either constructs a locally minimal tree of a given structure for a given boundary set, or reports that such a tree does not exist. This algorithm was discovered by Z. Melzak [9] and improved by F. Hwang [8]. Assume that we are given with a tree G whose vertex degrees are at most 3, a finite subset M of the plane, and a bijection ϕ : ∂G → M , where ∂G is the set of all vertices from G of degrees 1 and 2. To start with, partition the tree G into the union of so-called non-degenerate components Gi by cutting the tree at each its vertex of degree 2, see

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Figure 6: Tree G is partitioned into 4 non-degenerate components by cutting at vertices of degree 2. Figure 6. To construct locally minimal network Γ of type G spanning M in accordance with ϕ it suffices to construct each its component Γi of type Gi on the corresponding boundary Mi = ϕ(∂Gi ), where ∂Gi = ∂G ∩ Gi , in accordance with ϕi = ϕ|∂Gi and to verify the angles between the edges of the components at the vertices of degree 2. All these angles must be more than or equal to 120◦ , see Figure 6. Now we pass to the case of one non-degenerate component, i.e. we assume that G has no vertices of degree 2 and that ∂G consists of all the vertices of degree 1. Such trees are referred as binary. If |∂G| = 2, then the corresponding locally minimal tree Γ is a straight segment. Otherwise, it is easy to verify that each such tree G contains so-called moustaches, i.e. a pair of vertices of degree 1 neighboring with a common vertex of degree 3. Fix such moustaches {x, x0 } ⊂ ∂G, by y denote their common vertex of degree 3, and make a forward step of Melzak algorithm, see Figure 7, that reduces the number of boundary vertices by 1. Namely, we reconstruct the tree G by deleting the vertices x and x0 together with the edges xy and x0 y and adding y to the boundary of new binary tree; reconstruct the set M by deleting the points ϕ(x) and ϕ(x0 ) and adding a new point Axx0 which is the third vertex of a regular triangle constructed on the straight segment ϕ(x)ϕ(x0 ) in the plane; and reconstruct the mapping ϕ putting ϕ(y) = Axx0 . Notice that the point Axx0 can be constructed in two ways, because there are two such regular triangles. Thus, if the number of boundary vertices in the resulting tree is more than 2, then we can repeat the procedure described above. And

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Figure 7: A step of forward trace of Melzak algorithm.

if it becomes 2, then we can construct the corresponding locally minimal tree — the straight segment. Here the forward trace of Melzak algorithm stops. Now we have to reconstruct the initial tree, if possible. Thus, we have a straight segment I ⊂ R2 realizing locally minimal tree with unique edge ab, and at least one of its ending points has the form Axx0 , where x and x0 are the boundary vertices of the binary tree G from the previous step, neighboring with their common vertex of degree 3. Let this common vertex be a, that is a corresponds to Axx0 . We reconstruct G by adding edges ax and ax0 . Then we restore the points ϕ(x) and ϕ(x0 ) in the plane together with the regular triangle ϕ(x)ϕ(x0 )Axx0 , circumscribe the circle S 1 around it and consider the intersection of S 1 with the segment I, see Figure 8. If it does not contains a point lying at the smaller arc of S 1 restricted by ϕ(x) and ϕ(x0 ), then the tree G can not be reconstructed and we have to pass to another realization of the forward trace of the algorithm. Otherwise we put ϕ(a) be equal to this point. The straight segments ϕ(x)ϕ(a) and ϕ(x0 )ϕ(a) meet at ϕ(a) by 120◦ and together with the subsegment ϕ(a)ϕ(b) form a locally minimal binary tree Γ of type G with tree boundary vertices. We repeat this procedure until we either reconstruct the tree of type G, or verify all possible realizations of the forward trace and conclude that the tree of type G does not exists. The Melzak algorithm described above contains an exponential number of possibilities of its forward trace realization, due to two possible locations of each regular triangle constructed by the algorithm. This complexity can be reduced by means of modification suggested by F. Hwang [8]. He showed that considering a bit more complicated config-

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Figure 8: A step of back trace of Melzak algorithm.

Figure 9: Three binary trees with 4 vertices of degree 1.

urations of boundary points (four points corresponding to “neighboring moustaches” or three points corresponding to moustaches and “neighboring” degree-1 vertex) one always can understand which regular triangle must be chosen, see details in [8]. But unfortunately even a linear time realization of Melzak algorithm does not lead to a polynomial algorithm of a shortest tree finding. The reason is a huge number of possible structures of the tree G with |∂G| = n together with also exponential number of different mappings ϕ : ∂G → M for fixed ∂G and M . Even for binary trees we have 3 possibilities for n = 4, see Figure 9, and 15 possibilities for n = 5 (notice that the corresponding binary trees are isomorphic as graphs). For n = 6 we have two non-isomorphic binary trees and the number of possibilities becomes 90. It can be shown that the total number of possibilities can be estimated by Catalan number and grough exponentially. So, to obtain an efficient algorithms, we have to find some a priori restrictions on possible structures of minimal networks. In the next sub-

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Figure 10: A boundary set with 4 convexity levels. section we tell about the restrictions generated by geometry of boundary sets.

3.4

Boundaries Geometry and Networks Topology

Here we review our results from [10] and [7]. The goal is to find some restriction on the structure of locally minimal binary trees spanning a given boundary in the plane in terms of geometry of the boundary set. To do this we need to choose or to introduce some characteristics of the network structure and of the boundary geometry. As a characteristic of the geometry of a boundary set M we take the number of convexity levels c(M ). Recall the definition. Let M be a finite non-empty subset of the plane. Take the convex hull ch M of M and assign the points from M lying at the boundary of the polygon ch M to the first convexity level M (1) of M . If the set M \ M (1) is not empty, then define the second convexity level M (2) of M to be equal to the first convexity level of M \ M (1) , and so on. As a result, we obtain the partition of the set M into its convexity levels, and by c(M ) we denote the total number of this levels, see Figure 10. Now let us pass to definition of a characteristic describing the “complexity” of planar binary trees. Assume that we are given with a planar binary tree Γ, and let the orientation of the plane be fixed. For any its two edges, say e1 and e2 , we consider the unique path γ in Γ starting at e1 and finishing at e2 . All interior vertices of γ are the vertices of Γ having degree 3. Let us walk from e1 to e2 along γ. Then at each inte-

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Figure 11: Left and right turns in planar binary tree and its twisting number (left), and twisting number of locally minimal binary tree. rior vertex of γ we make either left, or right turn in Γ. Define the value tw(e1 , e2 ) to be equal to the difference between the numbers of left and right turns we have made. In other words, assign to an interior vertex of γ the label τ = ±1, where +1 corresponds to left turns and −1 to right turns. Then tw(e1 , e2 ) is the sum of these values, see Figure 11. Notice that tw(e1 , e2 ) = − tw(e2 , e1 ). At last, we put tw Γ = max tw(ei , ej ), where the maximum is taken over all ordered pairs of edges of Γ. If the tree Γ is locally minimal, then the twisting number between any pair of its edges has a simple geometrical interpretation, see Figure 11. Namely, since the angles between any neighboring edges are equal to 2π/3, then tw(ei , ej ) is equal to the total angle which the oriented edge rotates by passing from ei to ej , divided by π/3. It turns out, that the twisting number of a locally minimal binary tree with a given boundary is restricted from above by a linear function on the number of convexity levels of the boundary. Namely, the following result holds. Theorem 3.4 Let Γ be a locally minimal binary tree connecting the boundary set M that coincides with the set of vertices of degree 1 from Γ. Then
tw Γ ≤ 12 c(M ) − 1 + 5. The important particular case c(M ) = 1 corresponds to the vertex sets of convex polygons. Such boundaries are referred as convex.

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Theorem 3.5 Let Γ be a locally minimal binary tree with a convex boundary. Then tw Γ ≤ 5. Conversely, any planar binary tree Γ with tw Γ ≤ 5 is planar equivalent to a locally minimal binary tree with a convex boundary. Notice that the direct statement of Theorem 3.5 is rather easy to prove (it follows from the geometrical interpretation of the twisting number, easy remark that tw Γ always attains at boundary edges, and the monotony of convex polygonal lines). But the converse statement is quite non-trivial. The proof obtained in [7] is based on the complete description of binary trees with twisting number at most five, obtained in terms of so-called triangular tilings that will be discussed in the next subsection. Problem 3.6 Estimate the number of binary trees structures with n vertices of degree 1 and twisting number at most k. It is more or less clear that the number is exponential on n even for k = 5, but it is interesting to obtain an exact asymptotic.

3.5

Triangular Tilings and their Applications

It turns out that the description of planar binary trees with twisting number at most five can be effectively done in the language of planar triangulations of a special type which are referred as triangular tilings. The correspondence between diagonal triangulations of planar convex polygons and planar binary trees is well-known: the planar dual graph of such triangulation is a binary tree, see Figure 12, and each binary tree can be obtained in such a way. Here the vertices of the dual graph are centers of the triangles of the triangulation (medians intersection point) and middle points of the sides of the polygon; and edges are straight segments joining either the middle of a side with the center of the same triangle, or two centers of the triangles having a common side. In the context of locally minimal binary trees, the most effective way to represent the diagonal triangulations is to draw them consisting of regular triangles. Such special triangulations are referred as triangular tilings. The main advantage of the tilings is that the dual binary tree constructed as described above is a locally minimal binary tree with the corresponding boundary. Therefore, tilings “feel the geometry” of locally

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Figure 12: A diagonal triangulation and corresponding planar binary tree. minimal binary trees and turns out to be very useful in the description of such trees with small twisting numbers. The main difficulty in constructing a triangulation consisting of regular triangles for a given binary tree is that the resulting polygon can overlap itself. An example of such overlapping can be easily constructed from a binary tree Γ corresponding to the diagonal triangulation of a convex n-gon, n ≥ 6, all whose diagonals are incident to a common vertex. But the twisting number of such Γ is also at least 6. The following result is proved in [7]. Theorem 3.7 The triangular tiling corresponding to any planar binary tree with twisting number less than or equal to five has no self-intersections. Theorem 3.7 gives an opportunity to reduce the description of the planar binary trees with twisting number at most five to the description of the corresponding triangular tilings. To describe all the triangular tilings whose dual binary trees have the twisting number at most five, we decompose each such tiling into elementary “breaks”. The triangles of the tiling are referred as cells. A cell of a tiling T is said to be outer, if two its sides lie at the boundary of T considered as planar polygon. Further, a cell is said to be inner, if no one of its sides lies at the boundary, see Figure 13. An outer cell adjacent to (i.e. intersecting with by a common side) an inner cell is referred as a growth of T . A tiling can contains as un-paired growths, so as paired growths, see Figure 14.

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Figure 13: A triangular tiling with its dual binary tree (left), its outer cells (middle) and inner cells (right).

Figure 14: Un-paired and paired growths of a tiling (left), growths that should be deleted to get a skeleton (middle), corresponding decomposition into skeleton and growths (right).

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Figure 15: Branching points (left), linear parts (middle), and code of a skeleton (right). For each inner cell we delete exactly one growth adjacent to it, providing such growths exist. As a result, we obtain a decomposition of the initial tiling into its growths and its skeleton (a tiling without growths). Notice, that such a decomposition is not unique. It turns out that the skeletons of the tilings whose dual binary trees have twisting number at most five can be described easily. Also, the possible location of growthes in such tilings on their skeletons also can be described. The details can be found in [7] or [1]. Here we only formulate the skeletons describing Theorem and include several examples of its application. Inner cells of a skeleton S are organized into so-called branching points, see Figure 15. After the branching points deleting, the skeleton is partitioned into linear parts. Each linear part contains at most one outer cell. Construct a graph C(S) referred as the code of the skeleton S as follows: the vertex set of C(S) is the set of its branching points and of the outer cells of its linear parts. The edges correspond to the linear parts, see Figure 15. The following result is proved in [7]. Theorem 3.8 Consider all skeletons whose dual graphs twisting numbers are at most 5 and for each of these skeletons construct its code. Then, up to planar equivalence, we obtain all planar graphs with at most 6 vertices of degree 1 and without vertices of degree 2. In particular, every such skeleton contains at most 4 branching points and at most 9

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Figure 16: All possible codes of skeletons whose dual binary trees have twisting number at most 5. linear parts. All possible codes of such skeletons are depicted in Figure 16. This description of skeletons and corresponding tilings obtained in [7], was applied to the proof of inverse (non-trivial) statement of Theorem 3.5. In some sense, the proof obtained in [7] is constructive: for each tiling under consideration a corresponding locally minimal binary tree with a convex boundary is constructed. Another application is a description of all possible binary trees of the skeleton type that can be realized as locally minimal binary trees connecting the vertex set of a regular polygon. It turns out, see details in [1], that there are 2 infinite families of such trees and 1 finite family. The representatives of these networks together with the corresponding skeletons are shown in Figure 17.

4

Steiner Ratio

As we have already discussed in the previous Section, the problem of finding a shortest tree connecting a given boundary set is exponential even in two-dimensional Euclidean plane. On the other hand, in practice it is necessary to solve transportation problems of this kind for several thousands boundary points many times a day. Therefore, in practice some heuristical algorithms are used. One of the most popular heuristics for a shortest tree is corresponding minimal spanning tree. But using such

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Figure 17: Two infinite families of locally minimal binary trees connecting vertex sets of regular polygons, existing for any regular n-gon (left) and for 3k + 6-gons (middle), and the unique finite family, existing for 24-, 30-, 36- and 42-gons only (right). approximate solutions instead of exact one it is important to know the value of possible error appearing under the approximation. The Steiner ratio of a metric space is just the measure of maximal possible relative error for the approximation of a shortest tree by the corresponding minimal spanning tree.

4.1

Steiner Ratio of a Metric Space

Let M be a finite subset of a metric space (X, ρ), and assume that |M | ≥ 2. We put sr M = smt(M )/ mst(M ). Evidently, sr M ≤ 1. The next statement is also easy to prove. Assertion 4.1 For any metric space (X, ρ) and any its finite subset M ⊂ X, |M | ≥ 2, the inequality sr M > 1/2 is valid. Proof. Let G be a Steiner tree connecting M . Consider an arbitrary embedding of the graph G into the plane, walk around G in the plane and list consecutive paths forming this tour and joining consecutive boundary vertices from M . The length of each such path γP Q joining boundary vertices P Q, i.e. the sum of the lengthes of its edges, is more than or equal to the distance ρ(P, Q), due to the triangle inequality. Consider the cyclic path in the complete graph with vertex set M consisting of edges formed by the pairs of consecutive vertices from the tour, and let T be a spanning tree on M contained in this path. It is clear, that

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P ρ(T ) < (P,Q) ρ(γP Q ), where the summation is taken over all the pairs of consecutive vertices of the tour. On the otherPhand, each edge of the tree G belongs to exactly two such paths, hence (P,Q) ρ(γP Q ) = 2ρ(G). So, we have sr(M ) ≥ ρ(G)/ρ(T ) > 1/2. The Assertion is proved. The value sr(M ) is the relative error appearing under approximation of the length of a shortest tree for a given set M by the length of a minimal spanning tree. The Steiner ratio of a metric space (X, ρ) is defined as the value sr(X) = inf M ⊂X sr(M ), where the infimum is taken over all finite subsets M , |M | ≥ 2 of the metric space X. So, the Steiner ratio of X is the value of the relative error in the worse possible case. Corollary 4.2 For arbitrary metric space (X, ρ) the inequality 1/2 ≤ sr(X) ≤ 1 is valid. Exercise 4.3 Verify, that for any r ∈ [1/2, 1] there exists a metric space (X, ρ) with sr(X) = r, see corresponding examples in [2]. Sometimes, it is convenient to consider so-called Steiner ratios srn (X) of degree n, where n ≥ 2 is an integer, which are defined as follows: srn (X) = inf M ⊂X,|M |≤n sr(M ). Evidently, sr2 (X) = 1. It is also clear that sr(X) = inf n srn (X). Steiner ratio was firstly defined for the Euclidean plane in [11], and during the following years the problem of Steiner ratio calculation is one of the most attractive, interesting and difficult problems in geometrical optimization. A short review can be found in [2] and in [12]. One of the most famous stories here is connected with several attempts to prove so-called Gilbert–Pollack Conjecture, see [11], saying that sr(R2 , ρ2 ) = √ 3/2, where ρ2 stands for the Euclidean metric, and hence sr(R2 , ρ2 ) is attained at the vertex set of a regular triangle, see Figure 2. In 1990s D. Z. Du and F. K. Hwang announced that they proved the Steiner Ratio Gilbert–Pollak Conjecture [13], and their proof was published in Algorithmica [14]. In spite of the appealing ideas of the paper, the questions concerning the proof appeared just after the publication, because the text did not appear formal. And about 2003–2005 it becomes clear that the gaps in the D. Z. Du and F. K. Hwang work are too deep and can not be repaired, see detail in [15].

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Figure 18: To the calculation of sr3 (R2 , ρ2 ).

4.2

Steiner Ratio of Small Degrees for Euclidean Plane

Gilbert and Pollack calculated sr3 (R2 , ρ2 ) in their paper [11]. We include their proof here. √ Since the Steiner ratio of a regular triangle is equal to 3/2, then √ sr3 (R2 , ρ2 ) ≤ 3/2, so we just need to prove the opposite inequality. To do this, consider a triangle ABC in the plane. If one of its angles is more than or equal to 120◦ , then the shortest tree coincides with minimal spanning tree, so in this case sr(ABC) = 1. So it suffices to consider the case when all the angles of the triangle are less than 120◦ . Let S be the Torricelli point of the triangle ABC. Show firstly that |AS| ≤ |BS|, if and only if |BC| ≥ |AC|, i.e. the shortest edge of the Steiner minimal tree lies opposite with the longest side of the triangle. The proof is shown in Figure 18, left. Indeed, if |BS| < |AS|, then we take the point B 0 ∈ [S, B] with |SB 0 | = |SA|, hence |CB 0 | = |CA| due to symmetry and |CB 0 | < |CB| because B 0 ≥ 120c . Conversely, if |BC| > |B 0 C|, then there exists B 0 ∈ [B, S] with |CB 0 | = |CA|, because |BC| > |CA| > |SC|. Then |AS| = |SB 0 | < |SB|. Thus, the two-edges tree T = [A, B] ∪ [B, C] is a minimal spanning tree for ABC, if and only if BC is the longest side of ABC, if and only if |AS| ≤ |BS| and |AS| ≤ |CS|. Consider the points E ∈ [B, S] and D ∈ [C, S], such that |AS| = |ES| = |DS|, and put x = |AC|, y = |AB|, z = |DE| = |AD| = |AE|, and x0 = |CD|, y 0 = |EB|. Then

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√ |SA| = |SE| = |SD| = z/ 3 and smt(M ) = 3|SA|+|DC|+|EB| =

√

3z+x0 +y 0

and

mst(M ) = x+y,

where M stands for the set {A, B, C}. But x ≤ x0 + z and y ≤ y 0 + z, due to the triangle inequality, and hence √ √ √ √ 3z + x0 + y 0 3z + x0 + y 0 3z + x0 + y 0 3 sr(M ) = ≥ 0 = 0 ≥ . 0 0 x+y x +z+y +z x + y + 2z 2 Thus, we proved the following statement. Assertion 4.4 The following relation is valid: sr3 (R2 , ρ2 ) =

√

3/2. √ Remark. For small n it is already proved that srn (R2 , ρ2 ) = 3/2 (recently O. de Wet proved it for n ≤ 7, see [16]). The proof of de Wet is based on the analysis of Du and Hwand method from [14] and understanding that it works for boundary sets with n ≤ 7 points. Also in 60th several lower bounds for sr(R2 , ρ2 ) were obtained, and the best √ of them is worse than 3/2 in the third digit only. √ Problem 4.5 Very attractive problem is to prove that sr(R2 , ρ2 ) = 3/2, i.e. to prove Gilbert–Pollack Conjecture. The attempts to repair the proof of Du and Hwang have remained unsuccessful, so some fresh ideas are necessary here.

4.3

Steiner Ratio of Other Euclidean Spaces and Riemannian Manifolds

The following result is evident, but useful. Assertion 4.6 If Y is a subspace of a metric space X, i.e. the distance function on Y is the restriction of the distance function of X, then sr(Y ) ≥ sr(X). √ This implies, that sr(Rn , ρ2 ) ≤ sr(R2 , ρ2 ) ≤ 3/2. Recall that Gilbert–Pollack conjecture implies that the Steiner ratio of Euclidean plane attains at the vertex set of a regular triangle. In multidimensional case the situation is more complicated. The following result was obtained by Du and Smith [17]

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Figure 19: Construction of the set P in R3 (non-interesting but visual case n = 2). Assertion 4.7 If M ⊂ Rn is the vertex set of a regular n-dimensional simplex, then sr(M ) > sr(Rn , ρ2 ) for n ≥ 3. Proof. Consider the boundary set P in Rn+1 , consisting of the following 1 + n(n + 1) points: one point (0, . . . , 0) and n(n + 1) points all whose coordinates except two are zero, one is equal to 1, and the remaining one is −1. It is clear that P is a subset of n-dimensional plane defined by the next linear condition: sum of all coordinates is equal to zero. Represent P as the union of the subsets P i = {x ∈ P | xi = 1} ∪ {(0, . . . , 0)}. Notice that each set P i , i = 1, . . . , n + 1, consists of n + 1 points and forms the vertex set of an regular n-dimensional simplex (to see that it suffices to verify that all the distances between the pairs of points from √ P i are the same and are equal to 2). The configuration of 7 points in R3 is shown in Figure 19 (this case is not important for us, but it is easy to draw). Now, mst(P ) = (n + 1) mst(P i ), but for n ≥ 3 we conclude that smt(P ) < (n + 1) smt(P i ), because the degree of the vertex (0, . . . , 0) in the corresponding network which is the union of the shortest networks for P i is equal to n + 1 ≥ 4 that is impossible in the shortest network due to the Local Structure Theorem 3.1. So, sr(P ) = smt(P )/ mst(P )
1 is called the Steiner–Gromov ratio of the space X and is denoted by sgr(X ), or by sgr(X), if it is clear what particular metric on X is considered. Notice that sgrn (X ) is a non-increasing function on n. Besides, it is easy to see that sgr2 (X ) = 1 and sgr3 (X ) = 3/4 for any nontrivial metric space X . Assertion 5.19 The Steiner–Gromov ratio of an arbitrary metric space is not less than 1/2. There exist metric spaces whose Steiner–Gromov ratio equals to 1/2. Recently, A. Pakhomova, a student of Mechanical and Mathematical Faculty of Moscow State University, obtained an exact general estimate for the degree n Steiner–Gromov ratio, see [35]. Assertion 5.20 (A. Pakhomova) For any metric space X the estimate n sgrn (X ) ≥ 2n − 2 is valid. Moreover, this estimate is exact, i.e. for any n ≥ 3 there exists a metric space Xn such that sgrn (Xn ) = n/(2n − 2). Also recently, Z. Ovsyannikov [36] investigated the metric space of all compact subsets of Euclidean plane endowed with Hausdorff metric. Assertion 5.21 (Z. Ovsyannikov) The Steiner ratio and the Steiner– Gromov ratio of the metric space of all compact subsets of Euclidean plane endowed with Hausdorff metric are equal to 1/2.

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Steiner Subratio

Let X = (X, ρ) be an arbitrary metric space, and let M ⊂ X be some its finite subset. Recall that by smt(M, ρ) we denote the length of Steiner minimal tree joining M . Further, for nontrivial subsets M , we define the value ssr(M ) = mf(M, ρ)/ smt(M, ρ) and call it by the Steiner subratio of the set M . The value inf ssr(M ), where infimum is taken over all nontrivial finite subsets of X consisting of at most n > 1 points, is denoted by ssrn (X ) and is called the degree n Steiner subratio of the space X . At last, the value inf ssrn (X ), where the infimum is taken over all positive integers n > 1, is called the Steiner subratio of the space X and is denoted by ssr(X ), or by ssr(X), if it is clear what particular metric on X is considered. Notice that ssrn (X ) is a nonincreasing function on n. Besides, it is easy to see that ssr2 (X ) = 1 for any nontrivial metric space X . √ Proposition 5.22 ssr3 (Rn ) = 3/2. The next result is obtained by E. I. Filonenko, a student of Mechanical and Mathematical Department of Moscow State University, see [34]. √ Proposition 5.23 (E. I. Filonenko) ssr4 (R2 ) = 3/2. Conjecture 5.24 The Steiner subratio of the √ Euclidean plane is achieved at the regular triangle and, hence, is equal to 3/2. Recently, A. Pakhomova obtained an exact general estimate foe the degree n Steiner subratio, see [35]. Proposition 5.25 (A. Pakhomova) For any metric space {X } the estimate n ssrn (X ) ≥ 2n − 2 is valid. Moreover, this estimate is exact, i.e. for any n ≥ 3 there exists a metric space Xn such that ssrn (Xn ) = n/(2n − 2). Also recently, Z. Ovsyannikov [36] investigated the metric space of all compact subsets of Euclidean plane endowed with Hausdorff metric.

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Proposition 5.26 (Z. Ovsyannikov) Let C be the metric space of all compact subsets of Euclidean plane endowed with Hausdorff metric. Then ssr3 (C) = 3/4 and ssr4 (C) = 2/3.

References [1] A. O. Ivanov, and A. A. Tuzhilin. Branching Solutions to OneDimensional Variational Problems. Singapore, New Jersey, London, Hong Kong, 2001. 342 p. [2] A. O. Ivanov, and A. A. Tuzhilin. Extreme Networks Theory. Moscow, Izhevsk, 2003. 406 p. [In Russian.] [3] A. O. Ivanov, and A. A. Tuzhilin. One-dimensional Gromov minimal filling problem. // Sbornik: Mathematics. 2012. V. 203, No 5. P. 677726 [Matem. sb. 2012. T. 203, No 5. S. 65-118]. [4] V. A. Emelichev, at al. Lections on Graph Theory. Moscow, 1990. 384 p. [In Russian.] [5] B. Chazelle. The soft heap: an approximate priority queue with optimal error rate. // Journal of the Association for Computing Machinery. 2000. V. 47, No 6. P. 1012-1027. [6] A. O. Ivanov and A. A. Tuzhilin. The geometry of inner spanning trees for planar polygons. // Izvestiya: Mathematics. 2012. V. 76, No 2. P. 215–244 [Izv. RAN. Ser. Matem. 2012. T. 76, No 2. S. 3-36]. [7] A. O. Ivanov and A. A. Tuzhilin. The Steiner problem in the plane or in plane minimal nets.// Mathematics of the USSR-Sbornik. 1993. V. 74, No 2. P. 555–582 [Matem. Sb. 1991. T. 182, No 12. S. 18131844]. [8] F. K. Hwang. A linear time algorithm for full Steiner trees. // Oper. Res. Letter. 1986. V. 5. P. 235–237. [9] Z. A. Melzak. On the problem of Steiner. // Canad. Math. Bull. 1960. V. 4. P. 143–148.

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[10] A. O. Ivanov and A. A. Tuzhilin. Solution of the Steiner problem for convex boundaries. // Russian Mathematical Surveys. 1990. V. 45, No 2. P. 214-215 [Uspekhi Matem. Nauk. 1990. T. 45, No 2(272). S. 207-208]. [11] E. N. Gilbert and H..O. Pollak. “Steiner minimaltrees.” // SIAM J. Appl. Math. 1968. V. 16, No 1. P. 1–29. [12] D. Cieslik. The Steiner ratio. Springer, 2001. 256 p. [13] D. Z. Du, F. K. Hwang. The Steiner ratio conjecture of Gilbert– Pollak is true. // Proc. Nat. Acad. Sci. 1990. V. 87. P. 9464–9466. [14] D. Z. Du, F. K. Hwang. A proof of Gilbert–Pollak Conjecture on the Steiner ratio. // Algorithmica. 1992. V. 7. P. 121–135. [15] A. O. Ivanov and A. A. Tuzhilin. The Steiner Ratio GilbertPollak Conjecture Is Still Open. Clarification Statement. // Algorithmica. 2012. V. 62. No 1–2. P. 630–632. [16] P. O. de Wet. Geometric Steiner Minimal Trees. PhD Thesis. UNISA, Pretoria, 2008. [17] D. Z. Du and W. D. Smith. Disproofs of Generailzed Gilbert–Pollack conjecture on the Steiner ratio in three and more dimensions. // Combin. Theory. 1996. V. 74. Ser. A. P. 115–130. [18] W. D. Smith and J. M. Smith. On the Steiner ratio in 3-Space. // J. of Comb. Theory. 1995. V. 65. Ser. A. P. 301–322. [19] R. L. Graham and F. K. Hwang. A remark on Steiner minimal trees. // Bull. of the Inst. of Math. Ac. Sinica. 1976. V. 4. P. 177–182. [20] N. Innami and B. H. Kim. Steiner ratio for Hyperbolic surfaces. // Proc. Japan Acad. 2006. V. 82. Ser. A. No 6. P. 77–79. [21] D. Cieslik, A. Ivanov and A. Tuzhilin. Steiner Ratio for Manifolds. // Mat. Zametki. 2003. V. 74. No 3. P. 387-395 [Mathematical Notes. 2003. V. 74. No 3. P. 367-374]. [22] M. Gromov. Filling Riemannian manifolds. // J. Diff. Geom. 1983. V. 18. No 1. P. 1–147.

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[23] A. O. Ivanov, A. A. Tuzhilin. Steiner Ratio. The State of the Art. // Matemat. Voprosy Kibern. 2002. V. 11. P. 27–48 [In Russian]. [24] P A. Borodin. An example of nonexistence of a steiner point in a Banach space. // Matem. Zametki. 2010. V. 87. No 4. P. 514–518 [Mathematical Notes. 2010. V. 87. No 4. P. 485-488]. [25] A. O. Ivanov, Z. N. Ovsyannikov, N. P. Strelkova, A. A. Tuzhilin. One-dimensional minimal fillings with edges of negative weight. // Vestnik MGU, ser. Mat., Mekh. 2012. No 5. P. 3–8. [26] A. Yu. Eremin. Formula calculating the weight of minimal filling. // Matem. Sbornik. 2012. [To appear]. [27] Z. N. Ovsyannikov. Pseudo-additive metric spaces and minimal fillings. // Vestnik MGU. 2013. [To appear]. [28] K. A. Zareckii. Constructing a tree on the basis of a set of distances between the hanging vertices. // Uspehi Mat. Nauk. 1965. V. 20. No 6. P. 90-92. [In Russian]. [29] J. M. S. Sim˜ oes-Pereira. A note on the tree realizability of a distance matrix. // J. Combinatorial Th. 1969. V. 6. P. 303–310. [30] E. A. Smolenskij. About a Linear Denotation of Graphs. // J. vych. matem. i matem. phys. 1962. V. 2. No 2. P. 371–372. [In Russian]. [31] S. L. Hakimi, S. S. Yau. Distane matrix of a graph and its realizability. // Quart. Appl. Math. 1975. V. 12. P. 305–317. [32] O. V. Rubleva. Additivity Criterion for finite metric spaces and minimal fillings. // Vestnik MGU, ser. matem., mekh. 2012. No 2. P. 8– 11. [33] M. Gromov. Hyperbolic groups. // in book: S. M. Gersten, ed. Essays in Group Theory. 1987. Springer. [34] E. I. Filonenko. Degree 4 Steiner subratio of Euclidean plane. // Vestnik MGU, ser. matem., mekh. 2013. To appear. [35] A. S. Pakhomova. Estimates for Steiner–Gromov ratio and Steiner subratio. // Vestnik MGU, ser. matem., mekh. 2013. To appear.

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[36] Z. N. Ovsyannikov. Steiner ratio, Steiner–Gromov ratio and Steiner subratio for the metric space of all compact subsets of the Euclidean space with Hausdorff metric. // Vestnik MGU, ser. matem., mekh. 2013. To appear.

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The Borsuk–Ulam and ham sandwich theorems Roman Karasev

1

The Borsuk–Ulam theorem

We start with the classical Borsuk–Ulam theorem [1]: Theorem 1.1. For any continuous map f : S n → Rn there exists a pair of antipodal points x, −x ∈ S n such that f (x) = f (−x). Proof. By putting g(x) = f (x) − f (−x), we reduce this theorem to the following: For an odd map g : S n → Rn there exists a point x ∈ S n such that g(x) = 0. A map g is called odd if g(−x) = −g(x) for any x. Then we consider a simple map g0 defined as follows: If S n is the unit sphere in Rn+1 , then g0 is the projection to a coordinate subspace Rn ⊂ Rn+1 . It is easy to observe that for g0 there is a unique antipodal pair x0 , −x0 ∈ S n that is mapped to zero. Moreover, at this x0 (and −x0 ) the Jacobian matrix Dg0 is nondegenerate. Assume that g does not map any point to zero. Now we connect g0 and g by the homotopy ht (x) = (1 − t)g0 (x) + tg(x). For any t the map ht (x) remains an odd continuous map. From standard facts of differential geometry we may perturb the homotopy ht slightly to obtain ˜ t (x) with the following properties: another homotopy h 1) h0 (x) is still equal to g0 (x); 2) zero is a regular value for h : S n × I → Rn (I = [0, 1] is the segment), and h−1 (0) is a one-dimensional submanifold Z ⊂ S n × I with boundary in S n × ∂I. 3) the map h1 (x) may be not equal to g(x), but it still misses zero in Rn . Now starting from the unique pair {(x0 , 0), (−x0 , 0)} ∈ ∂Z and trace this pair along the one-dimensional set Z. This pair of point must finally arrive at some other pair {(x1 , t1 ), (−x1 , t1 )} ⊂ S n × I, but there is nowhere to arrive: t1 = 1 is impossible because of the assumption (3), t1 = 0 would mean that the pair (x1 , −x1 ) is the same as (x0 , −x0 ) but with reversed order. The latter is impossible because if (x0 , 0) and (−x0 , 0) are connected by a component of Z then the antipodal action 111

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(x, t) 7→ (−x, t) would have a fixed point in this component, which is wrong. Thus the assumption was wrong and we conclude that g −1 (0) is nonempty. Let us state another similar theorem: Theorem 1.2. Any odd map g : S n → S n has odd degree. Proof. The proof follows from taking quotient by the antipodal action RP n = S n /Z2 and considering the induced map g 0 : RP n → RP n . One may check that the map g∗0 : H1 (RP n ) → H1 (RP n ) is an isomorphism. Then from the explicit description of the cohomology H ∗ (RP n ; F2 ) = F2 [w]/(wn+1 ) it follows that g 0 induces an isomorphism in modulo 2 cohomology and therefore its degree is odd. The above theorem has the following corollary due to H. Hopf [4]: Theorem 1.3. Let M be a compact n-dimensional Riemannian manifold and δ > 0 is a positive real number. For any map f : M → Rn there exist two points x, y ∈ M connected by a geodesic of length δ such that f (x) = f (y). The proof is left to the reader. Hint: Consider the point x ∈ M such that f (x) is the extremal point of the image f (M ). Then for every direction ν ∈ Tx M consider the geodesic `(t, ν) from x in the direction of ν and, assuming the contrary, construct two homotopic maps from the set of directions (identified with S n−1 ) to S n−1 , one of them being odd and the other being non-surjective (and therefore having zero degree). For more information about the Borsuk–Ulam theorem the reader is referred to the book of Matouˇsek [6].

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The ham sandwich theorem and its polynomial version

Now we are ready to prove the classical ‘ham sandwich’ theorem [9, 8]: Theorem 2.1. Let µ1 , . . . , µn be probability measures in Rn that attain zero on every hyperplane. Then some hyperplane H partitions Rn into a pair of halfspaces H + and H − so that µi (H + ) = µi (H − ) = 1/2 for any i. Proof. We put A = Rn to Rn+1 as the affine hyperplane defined by xn+1 = 1. Then for any unit vector ν ∈ S n the inequality (ν, y) ≥ 0 defines a halfspace Hν+ in A with the complement Hν− . For ν equal to (0, . . . , 0, ±1) those halfspaces become degenerate, that is coinciding with the empty set of with the whole A. Now we consider the map f : S n → Rn defined as follows: f (ν) = (µ1 (Hν+ ), . . . , µ1 (Hν+ )). + )= Be Theorem 1.1 there exist a pair ν, −ν ∈ S n with µi (Hν+ ) = µi (H−ν − µi (Hν ) for any i. Since the total measure of A is 1 with respect to each µi , we obtain µi (Hν+ ) = µi (Hν− ) = 1/2 for any i.

Now we are going to consider more general partitions of the space. We start from the line R and consider the space of polynomials of degree at most d, which we denote by Pd (R). For every f ∈ Pd (R) it is natural to consider the sets Hf+ = {x : f (x) ≥ 0}

and Hf− = {x : f (x) ≤ 0}.

We claim that for any d absolute continuous probability measures µ1 , . . . , µd in R there exists a polynomial f ∈ Pd (R) that splits (with R = Hf+ ∪Hf− ) every measure into two equal halves. This fact is established by considering the moment map v1d : R → Rd that takes t to the vector (t, t2 , . . . , td ). The images of the measures µi are defined and it is important that they attain zero in every halfspace; this follows from the fact that the original µi attain zero on every finite set. Now we apply the ham sandwich theorem to these measures in Rd and obtain an equipartitioning halfspace in Rd with equation λ(x) ≥ 0,

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where λ is a linear function with possible constant term. The functions λ(v1d ) then becomes a polynomial of degree at most d in one variable. A nontrivial generalization of this one-dimensional fact for splitting into not equal parts is given in [10]. The general case of this results follows by considering the Veronese d+n map vnd : Rn → R( n ) − 1 that takes an n-tuple (x1 , . . . , xn ) to the set of all possible nonconstant monomials in xi ’s of degree at most d. Therefore we obtain:
Theorem 2.2. Let n and d be positive integers and r = d+n −1. Then n any r absolutely continuous measures µ1 , . . . , µr in Rn may be partitioned into equal halves simultaneously by a partition Rn = Hf+ ∪ Hf− , where f is a polynomial of degree at most d. A similar result holds for finite point sets.
Theorem 2.3. Let n and d be positive integers and r = d+n − 1. n Then for any r finite sets X1 , . . . , Xr in Rn there exists a partition Rn = Hf+ ∪ Hf− , where f is a polynomial of degree at most d, such that |Xi ∩ Hf+ |, |Xi ∩ Hf− | ≥ 1/2|Xi | for any i. Proof. Replace every point x ∈ Xi with a density distributed uniformly over a ball Bε (x) and sum those densities over all x ∈ Xi to obtain the density of the measure µi . Then apply Theorem 2.2 to µi and pass to the limit ε → +0. It is easy to see that all possible partitioning polynomials fε may be chosen to be contained in a bounded subset of Pd (Rn ) and therefore it is possible to select a limit polynomial f that will satisfy the requirements.

3

Partitioning a single point set with successive polynomials cuts

In the review of Kaplan, Matouˇsek, and Sharir [5] the importance of the following corollary of the polynomial ham sandwich theorem is emphasized: Lemma 3.1. Let X be a finite set in Rn and r be a positive integer. It is possible to find a polynomial of degree at most Cn r1/n with the following

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property: The set Z = {x : f (x) = 0} partitions Rn into connected components V1 , . . . , VN so that |X ∩ Vi | ≤ 1/r|X| for every i. Proof. We first use Theorem 2.3 to partition X into almost equal halves using the zero set Zf1 of a linear function f1 . Then we partition every part into equal halves with another zero set Zf2 of a function f2 , which may be still chosen to be linear if n ≥ 2. Then we do the same j times. After that, we have a collections of polynomials f1 , . . . , fj and consider their product f = f1 f2 . . . fj . The zero set Zf partitions Rn into at least r = 2j connected components, each containing at most 1/r fraction of the set X. It remains to bound from above the degree of f . On the i-th step we partitioned 2i−1 sets and the required degree of the polynomials was at most (n!2i−1 )1/n . The summation over i of this geometric progression gives the estimate deg f ≤

(n!r)1/n = Cn r1/n . 1 − 2−1/n

We proved the result for r powers of two, for other r we could choose 2j to be the least power of two not less than r. Following [5], we make several comments on this lemma. Seemingly we partitioned the space into 2j parts, but some parts could actually split into several connected component in that process. So we actually do not control the number of parts. The other issue is that some points of X (and actually many of them) can lie on the set Zf and need a separate treatment in most applications.

4

The Szemer´ edi–Trotter theorem

We are going to apply Lemma 3.1 and deduce the Szemer´edi–Trotter theorem. We start from the definition: Definition 4.1. Let P be a set of points and L be a set of lines in the plane. Denote by I(P, L) their incidence number, that is the number of pairs (p, `) ∈ P × L such that p ∈ `. Theorem 4.2. In the plane I(P, L) ≤ C(|P |2/3 |L|2/3 + |P | + |L|) for a suitable absolute constant C.

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Remark 4.3. This theorem is also valid for pseudolines, that is subsets of the place that behave like lines in terms of their intersection. Such a generalization so far seems to be out of reach of algebraic methods. Proof. We start from a much weaker estimate, which we are going to apply to different sets of points and lines: Lemma 4.4. I(P, L) ≤ |L| + |P |2 . This lemma is proved by splitting L into two families: one for the lines intersecting at most one point of P and all other lines. The details are left to the reader. Now we put m = |P | and n = |L|. Take some parameter r, whose √ value we will define later. We choose an algebraic set Z of degree O( r) (from now on we use the notation O(·) to avoid different constants) that partitions R2 into connected sets V1 , . . . , VN . Let P0 = P ∩ Z and Pi = P ∩ Vi for any i. Also denote by L0 the lines in L that lie entirely on Z and denote by Li the lines in L √ that intersect Vi . Note that Li ’s are not disjoint, and |L0 | ≤ deg Z = O( r). The √ crucial fact is that every line from √L \ L0 intersects Z in at most O( r) points and intersects at most O( r) of the regions Vi . First, we obviously estimate: √ I(P0 , L0 ) ≤ m|L0 | = O(m r),

X

√ I(P0 , Li ) = nO( r),

I(Pi , L0 ) = 0.

i

√ Summing up those obvious estimates we obtain O((m + n) r) in total. It remains to use Lemma 4.4 and bound X X √ I(Pi , Li ) ≤ |Li | + |Pi |2 ≤ nO( r) + m2 /r. i

i

Now we make several observations. The projective duality allows us to interchange points and lines and assume m ≤ n. Then Lemma 4.4 √ allows us to concentrate on the case n ≤ m ≤ n. After that putting 4/3 r = m we make all the estimates made so far to be of the form n2/3 2/3 2/3 O(n m ).

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Spanning trees with low crossing number

Another application of Lemma 3.1 is the following theorem of Chazelle and Welzl [12, 2, 13]: Theorem 5.1. Any finite set P ⊂ R2 has a spanning tree T with the following property: Any line p ` (apart from a finite number of exceptions) intersects T in at most C |P | points. Proof. The first observation is that it is sufficient to find an arcwise connected subset X containing P and having small crossings with almost all lines. Then it is easy to select a tree T inside X that will still connect P and has crossings at most twice of the crossings of X. Then the edges of T may be replaced with straight line segments without increasing the number of crossings. The details of this reduction are left to the reader. Now we prove the following: Lemma 5.2. It is possible to find a set Y ⊃ P with atpmost |P |/2 connected components and line crossing number at most C |P |. The lemma is proved as follows: Taking r = |Pp|/C1 we obtain by Lemma 3.1 an algebraic set Z of degree at most C2 |P |/C1 that splits P into parts of size at most C1 . The number of connected components
of Z is bounded from above by the number deg 2Z−1 + 1 by the Harnack theorem [3]1 . Hence for sufficiently large C1 (but still an absolute constant) the number of connected components of Z will be at most |P |/2. Then in every component Vi of the complement R2 \ Z we have at most C1 points of P , which we span by a tree Ti and attach this tree to the set Z. Put Y = Z ∪ T1 ∪ · · · ∪ TN . Any line p ` (apart from a finite number of exceptions) will intersect Z at most |P | times and will intersect p p at most |P | trees of Ti . Hence this line will have at most (1 + C1 ) |P | points of intersection with Y . The lemma is proved. Now we apply the lemma once, then select a point in every component of Y thus obtaining the set P2 with |P2 | ≤ 1/2|P |. Then apply the lemma again to P2 , pass to another point set P3 with |P3 | ≤ 1/2|P2 | and so on. As it was in the proof of Lemma 3.1, in log |P | number of steps we 1 The reader may try to prove the Harnack theorem considering the real algebraic curve as
a set of cycles on the corresponding complex algebraic curve of genus g = deg Z−1 . 2

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arrive at a connected set X = Y1 ∪ Y2 ∪ . . . spanning P . The number of crossings of X with a line ` is bounded from above by the sum of a geometric 2−1/2 and the leading term p p progression with denominator (1 + C1 ) |P |, so it is bounded by C |P |, where C is another absolute constant. The reader is now referred to the review [5] and a more advanced paper of Solymosi and Tao [7] for other interesting applications of Lemma 3.1.

References [1] K. Borsuk. Drei S¨ atze u ¨ber die n-dimensionale euklidische Sph¨are. Fundam. Math, 20(1):177–190, 1933. [2] B. Chazelle and E. Welzl. Quasi-optimal range searching in spaces of finite VC-dimension. Discrete Comput. Geom., 4:467–489, 1989. ¨ [3] C. G. A. Harnack. Uber Vieltheiligkeit der ebenen algebraischen Curven. Math. Ann., 10:189–199, 1876. [4] H. Hopf. Eine Verallgemeinerung bekannter Abbildungs und ¨ Uberdeckungss¨ atze. Portugaliae Math., 4:129–139, 1944. [5] H. Kaplan, J. Matouˇsek, M. Sharir. Simple proofs of classical theorems in discrete geometry via the Guth–Katz polynomial partitioning technique. Discrete Comput. Geom., 48(3):499–517, 2012. [6] J. Matouˇsek. Using the Borsuk-Ulam theorem: lectures on topological methods in combinatorics and geometry. Springer Verlag, 2003. [7] J. Solymosi and T. Tao. An incidence theorem in higher dimensions. Discrete Comput. Geom., 48(2):255–280, 2012. [8] H. Steinhaus. Sur la division des ensembles de ´espace par les plans et des ensembles plans par les cercles. Fundam. Math, 33:245–263, 1945. [9] A. H. Stone and J. W. Tukey. Generalized “sandwich” theorems. Duke Mathematical Journal, 9(2):356–359, 1942.

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[10] W. Stromquist and D. Woodall. Sets on which several measures agree. J. Math. Anal. Appl., 108:241–248, 1985. [11] E. Szemer´edi and W. T. Trotter. Extremal problems in discrete geometry. Combinatorica, 3:381–392, 1983. [12] E. Welzl Partition trees for triangle counting and other range searching problems. Proc. 4th Annu. ACM Sympos. Comput. Geom., 23– 33, 1988. [13] E. Welzl On spanning trees with low crossing numbers. Data structures and efficient algorithms, Lecture Notes in Computer Science, vol. 594. Springer Verlag, 1992, 233–249.

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Robust Geometric Computation M. Kerber

Geometric algorithms are often composed of a set of basic geometric predicates and constructions, also called primitives. Correctness of an algorithm is usually proved under the assumption that all primitives return the correct result. This is in sharp contrast to the naive, but common strategy of implementing such primitives with fixed-precision floating-point arithmetic. The disastrous effects of this gap between theory and practice have been witnessed in various areas using geometric algorithms, like computer-aided design or computational fluid dynamics. This mini-course is intended to raise the awareness of the described (non-)robustness problem and presents several principles to solve the problem.

1. The disasters of floating point arithmetic We demonstrate by a simple example the unexpected outcome of geometric algorithms when implemented using floating-point arithmetic. The content of this lecture is a combination of the very accessible paper by Kettner et al. [1] and the first chapter of the book manuscript by Mehlhorn and Yap [2]. A convex hull algorithm Our running example in this lecture is an incremental algorithm for computing the convex hull of a point set. A set C ⊆ Rd is convex if for any a, b ∈ C, the connecting line segment ab is also completely contained in C. For any set S ⊆ Rd , the convex hull of S, chS is the smallest convex set that contains S (equivalently, the intersection of all convex sets that contain S). If S is convex, chS = S. We concentrate on the case that S is a finite set of points in R2 ; in this 120

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case, chS is a simple polygon; all vertices of the polygons are points in S; in other words, S decomposes into convex hull points and interior points. We consider the problem of computing the convex hull of a set of points in R2 ; see Figure 1 for an illustration.

Abbildung 1. The convex hull of a point set (shaded area). The points denoted as circles are convex hull vertices, the points denoted as squares are interior. For three points p = (px , py ), q = (qx , qy ), and r = (rx , ry ), let pqr denotes the path consisting of two line segments, first going from p to q and then going from q to r. Also, call p, q, r collinear if they lie on a common line in R2 . To formulate our algorithm, we need the orientation of the triple (p, q, r) as follows (see Figure 2):   +1 if pqr is a left turn orient(p, q, r) := −1 if pqr is a right turn   0 if p, q, r are collinear. It can be shown by elementary arguments that 1 px py orient(p, q, r) = sign 1 qx qy 1 rx ry =



sign ((qx − px )(ry − py ) − (qy − py )(rx − px )) .

The orientation function above is the first example of a geometric primitive, which is a function defined in the coordinates of the input points.

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r q

p

Abbildung 2. An example where orient(p, q, r) = −1. More generally, every point r0 on the right of dashed line satisfies orient(p, q, r0 ) = −1, on the left of the dashed line orient(p, q, r0 ) = +1 and on the dashed line, orient(p, q, r0 ) = 0. Two types of primitives are distinguished: Geometric predicates which return either +1, −1 (representing the values True and False as in usual predicates), or return 0 in degenerate cases, such as collinearity of three points. Of course, orient is an example of a geometric predicate. Geometric constructions return another geometric object such as a point or line segment. We will see several additional geometric primitives as the course goes on. We say that a point r sees the line segment pq is orient(p, q, r) = −1. In this case, we also say that pq is visible from r. pq is weakly visible from r if orient(p, q, r) ≤ 0. We represent a convex polygon by a list of points (v0 , . . . , vk ), meaning that the polygon consists of the directed line segments v0 v1 , v1 v2 , . . . , vk−1 vk , vk v0 which are arranged in counterclockwise order. The following property is the key for our incremental convex hull algorithm (see also Figure 3): Proposition. Let P be a convex polygon represented by (v0 , . . . , vk ). A point r is outside of P if and only if it sees some edge of P . Moreover, the edges that are weakly visible from r form a connected subchain. We describe the convex hull algorithm now. The input are points s1 , . . . , sn in R2 , the output is a list L = (`0 , . . . , `k ) representing the convex hull polygon. The algorithm first chooses three points from s1 , s2 , s3 from S and initializes L as the counterclockwisely arranged triangle for-

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F D

C

E A

B

Abbildung 3. Consider the polygon represented by (A, B, C, D). Point E does not see any edge of the polygon (indeed, it is on the left of each edge). Point F , however, sees edges CD and DA. To update the convex hull, we can simply replace these two edges by the edges CF and F A. med by s1 , s2 , s3 . Then, it traverses the remaining points in S one by one. For every point r, it computes the subchain of segments that is visible from r. If there is no visible segment, s is in the interior. Otherwise Let this subchain start at `i and end at `j . Then, all points `i+1 , . . . , `j−1 are removed from L and r is inserted between `i and `j , and the algorithm proceeds with the next point. We remark that the described algorithm has a quadratic running time in the number of input points. There are more efficient algorithms that compute the convex hull in O(n log n) time. What makes the algorithm attractive for us is that it only uses a single geometric primitive, the orientation test. Double arithmetic Doubles are a standard number type that is efficiently realized in hardware. Formally, a double is a number of the form ±m2e with m = 1.m1 m2 . . . m52 , mi ∈ {0, 1} and e ∈ {−1023, . . . , 1023}. m and e are called mantissa and exponent, respectively. Arithmetic operations like additions and multiplications are rounded to the nearest number representable by a double by default. We perform a small experiment: Set p = (0.5, 0.5), q = (12, 12), and r = (24, 24). For 0 ≤ X, Y ≤ 255, we compute orient(p + 2−53 ·

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(X, Y ), q, r) using double arithmetic. We give each of the three possible results a different color and draw the result as a (256 × 256) image. See Figure 4. We observe that the wrong result is computed even considerably far away from the diagonal. Perhaps even worse, we observe inversions in the result, that is, the double-arithmetic version sometimes returns −1 where +1 is the exact result.

Abbildung 4. On the left: the exact result of orient. On the right: Result when using double arithmetic.

Disaster on microscopic scale We demonstrate the consequences of the inaccurate orientation predicate for the presented convex hull algorithm. It will turn out that the perhaps intuitive answer that a “slightly wrong” answer is computed is totally mistaken. p7

p7

p8 p2 , p3

p6

p2 , p3 p9

p6

p9

p1

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p5 p4

Abbildung 5. An input for the convex hull algorithm, and the computed result using double arithmetic.

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Consider the first example in Figure 5. It is chosen such that p2 and p3 are very close to each other, and lie almost on the line defined by p1 and p4 . The points are processed in the algorithm in the order p1 , p2 , . . . , p9 . By our choices, p4 is considered to be inside the triangle p1 p2 p3 , because the orientation test in double arithmetic (wrongly) yields the same result for p1 , p2 , p4 , p1 , p3 , p4 and p2 , p3 , p4 . Although the algorithm evaluates all further orientations correctly, it does not recover from not putting p4 into the convex hull. A second example is depicted in Figure 6. Again, we choose a pair of close-by points, p1 and p5 , which are nearly collinear with p4 and p6 . The algorithm computes the correct convex hull for p1 , p2 , p3 , p4 , but considers p5 to be outside that convex hull, because p5 sees p4 , p1 . This yields the intermediate “convex” hull (p1 , p2 , p3 , p4 , p5 ) which is not really convex. Continuing with p6 , the algorithm computes that it sees p4 p5 and p1 p2 , but not p5 p1 (note that this already contradicts the previous proposition that the visible segments should be connected). However, the algorithm has two choices where to add p6 ; see Figure 6 for both results. Note that of them yields a polygon with self-intersection which is also far away from being convex. Disasters on macroscopic scale The effect of rounding errors and inconsistent output is not just of academic interest – commercial software systems suffer from that principle, even for relatively simple tasks. CAD (computer-aided design) systems are used for creation, modelling, analyzing and optimizing geometric shapes. Many commercial software packages exist. A standard operation contained in most of them are Boolean set operations like union, intersection, or symmetric difference of two shapes. How robust is this operation? We test by taking the union of a regular n-gon and the rotation of the n-gon around its center by some angle α. The resulting shape is a 4n-gon for almost all choices of α. Table 1 shows the outcome for several commercial CAD systems. The rounding errors have different effects on the execution: in some cases, “just” the wrong result is computed. In other cases, the inconsistent intermediate results cause the algorithm to enter a infinite loop or to crash. Meshing software is another area that suffers from non-robustness. Mehlhorn and Yap quote two experts in computational fluid dynamics (CFD) as follows:

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p4

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p2

p1 p5

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Abbildung 6. Top: The situation of inserting the points p1 , . . . , p5 . Bottom: Possible outcomes after inserting p6 .

In a typical CFD aircraft analysis with 50 million elements, we spend 10-20 minutes for surface mesh generation, 3-4 hours for volume meshing, 1 hour for the actual flow analysis, and finally 2-4 weeks for debugging and geometry repair.

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Robust Geometric Computation SYSTEM ACIS ACIS ACIS Microstation95 Microstation95 Rhino3D Rhino3D

n 1000 1000 1000 100 100 200 400

127 α 1.0e-4 1.0e-5 1.0e-6 1.0e-2 0.5e-2 1.0e-2 1.0e-2

TIME 5 min 4.5 min 30 min 2 sec 3 sec 15 sec -

OUTPUT correct correct too difficult! correct incorrect! correct crash!

Tabelle 1. Results of the several commercial CAD systems for boolean operations (table is taken from the manuscript by Mehlhorn/Yap). For α, a value of 1.0e − 4 denotes the double with mantissa 1.0 and exponent −4. Some (non)-responses to Nonrobustness The following responses are common when people are facing the just presented nonrobustness problem Nonrobustness cannot be avoided Because computers can only store numbers with a finite representation, it is indeed impossible to get rid of rounding errors in general. But this does not mean that algorithms have to suffer from numerical inaccuracies as much as presented in this lecture. Nonrobustness only happens in ill-conditioned cases and can therefore be solved by avoiding such instances It is a fact that nonrobust implementations of geometric algorithms work for most inputs. However, bad inputs can happen with non-negligible frequency, either just by chance (and because input sizes become larger and larger, the probability of such events increases) or such degenerate inputs are forced by design (for instance, collinearity events are likely to happen when modelling shapes that actually contain a line) Nonrobustness can easily be solved by “epsilon-tweaking” Many programmers try to solve the problem by slightly changing the geometric predicate such that it returns zero if the determinantal expression has an absolute value smaller than some threshold . In other words, three points almost being collinear are considered to be collinear. It is true that such an approach eliminate some sources of errors (for instance, inversions in the predicates, as observed above, cannot happen anymore). On

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the other hand, it also complicates matters, as one formally deals with -thickened versions of points and lines. This complicates constructions like computing the intersection of two lines. In general, it is still a nontrivial task to turn a epsilon-tweaked version of geometric algorithms robust. Outline: In the following three lectures, we will discuss three different approaches to tackle the nonrobustness problem. • Degree-driven algorithmic design • Controlled perturbation • Exact Geometric Computation

2. Degree-driven Algorithm Design A relatively simple way to avoid rounding errors is to perform internal computations with a sufficiently high precision. However what does “sufficient” mean in this context? We consider a formalization that investigates the degree of an algorithm to determine a sufficient working precision. The content of this lecture is a simplified version of a paper by Liotta, Preparata and Tamassia [3]. The degree of an algorithm Recall that a geometric primitive P is a function in the coordinates of the input points. Let p1 , . . . , pk denote a set of k input points (with k some constant) with coordinates pi = (xi , yi ). We say that P is of degree d if P (p1 , . . . , pk ) is determined by a constantly many expressions of the form signf (x1 , . . . , xk , y1 , . . . , yk ) where f is a polynomial of degree at most d in 2k variables. The orientation predicate from Lecture 1 of degree 2. We consider several further predicates and analyze their degree.

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We consider the lexicographic comparison of two points  +1 if x1 < x2 or     (x1 = x2 and y1 < y2 ) lex_compare(p1 , p2 ) :=  0 if p = q.    −1 otherwise Clearly, lex_compare is determined by the signs of the two polynomials x1 − x2 and y1 − y2 and hence of degree 1. The next predicates compares the distance of a point to two others:   +1 if kq − p1 k < kq − p2 k is_closer_to(q, p1 , p2 ) := 0 if kq − p1 k = kq − p2 k   −1 if kq − p1 k > kq − p2 k Clearly, the value of the predicates depends on the expression kq − p1 k − kq − p2 k, but it contains square roots and is therefore not a polynomial. However, as only the sign matters, we can simply compare the squared distance function:
is_closer_to(q, p1 , p2 ) = sign kq − p1 k2 − kq − p2 k2 which is of degree 2. For the final example, note that three points p1 , p2 , p3 in the plane define a unique circle, defined by C(p1 , p2 , p3 ) (this circle degenerates into a halfplane if the three points are collinear). The following predicate determines the position of a fourth point with respect to that circle.   +1 if q lies inside C(p1 , p2 , p3 ) in_circle(q, p1 , p2 , p3 ) := 0 if q lies on C(p1 , p2 , p3 )   −1 if q lies outside C(p1 , p2 , p3 ) This predicate can be expressed by sion  kp1 k  kp2 k sign   kp3 k kqk

the following determinantal expres x1 y1 1 x2 y2 1   x3 y3 1  xq yq 1

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with q = (xq , yq ). Expanding the determinant, this yields a polynomial of degree 4. We define an algorithm to be of degree d if all the primitives that it uses are of degree d or less. For instance, the convex hull algorithm from Lecture 1 is of degree 2 because it uses only the orientation test. The definition is useful because of the following fact: If an algorithm is of degree d, all input points are of precision p, and all computations are performed with a working precision of d · (p + O(1)), all predicates return the correct result. The O(1) factor above means some constant that does not depend on d or p. When following the approach of choosing a sufficient working precision, the degree of an algorithm becomes another important measure for the quality of an algorithm besides time and space complexity. Closest site queries We consider the following simple problem: Given n points p1 , . . . , pn , usually called sites and a query point q in the plane, we want to find i ∈ {1, . . . n} such that pi is closest to q, that means, no pj with j 6= i is closer to q than pi . It is obvious how to solve this problem: We initialize i ← 1 and traverse the sites p2 , . . . , pn . If some pj is closer to q than pi , we set i ← j and continue. At the end, i is the index of a closest site. The algorithm has a running time proportional to n and is of degree 2 because it requires n − 1 executions of the predicate is_closer_to. A more interesting variant of the problem is to fix the sites p1 , . . . , pn , but to consider many query points (which might be unknown in the beginning). In this case, it makes sense to preprocess the sites p1 , . . . , pn in order to answer closest site queries more efficiently. The Voronoi diagram is the planar subdivision of the plane into vertices, edges, and faces such that all points in a face have a unique closest site, all points on an edge have exactly two closest sites and a vertex has three or more closest sites. See Figure 7 for an illustration. Note that every Voronoi edge is part of the bisector of two sites that are closest to the edge. A Voronoi vertex is a point where (at least) three bisectors meet. An equivalent description is that a Voronoi vertex is the center of a circle spanned by three sites such that no other site is in the interior of the circle. This property shows the relation of Voronoi diagrams with the in_circle predicate described before. Indeed, the

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2 v

1

w

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4

Abbildung 7. The Voronoi diagram of four points. The diagram contains the Voronoi vertices v which has the sites 1,2,4 as closest sites, and the Voronoi vertex w which has 2,3,4 as closest sites. The points on the Voronoi edge connecting v and w have 2 and 4 as closest site. Voronoi diagram of a set of n sites can be computed in O(n log n) time using O(n) space, and the algorithm is of degree 4. Assume that a Voronoi diagram for a set of sites p1 , . . . , pn is given such that each Voronoi face, edge and vertex knows the set of its closest sites. Given a query point q, we “just” have to find the Voronoi cell which it belongs to. This is a well-known problem denoted as point location. We have to transform the Voronoi diagram in a more sophisticated data structure to support point location queries. Several efficient solutions exist; we just use the most naive one for the sake of simplicity. Decompose the Voronoi diagram into vertical slabs. That means, at any Voronoi vertex, we simply shoot a ray upwards and downwards and split any Voronoi edge hit by the ray into two parts. The resulting planar subdivision of the plane is called slab decomposition. For any slab, we maintain an ordered list of Voronoi edges of the original diagram that passes through the slab. See Figure 8. Using a slab decomposition, we can answer closest site queries in the following way: given a query point q, we first determine the slab that q belongs to. This can be done by binary search on the x-coordinates of Voronoi vertices. The underlying predicate is to compare the x-coordinates of q with the x-coordinates of a Voronoi vertex. Once the slab is determined, another binary search is performed to determine the Voronoi edges just above and below q within the slab. The underlying predicate is to check whether q is above or below some Voronoi edge. Since the

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3 8

5 4

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Abbildung 8. The slab decomposition of a planar subdivision. The newly inserted vertices are drawn as little squares. Note that the vertical order of segments is unique within one slab. For instance, the shaded slab stores the segments in the order 26, 46, 47, 57, 58. number of slabs and the number of edges within one slab is at most n, this strategy answers a closest site query in O(log n) time. Analysis We analyze the degree of the just proposed method. For that, we assume from now on that the query point is given in the same precision as the sites p1 , . . . , pn . We have two predicates involved in the method. The first one is to compare x-coordinates of the query point and a Voronoi vertex v. Recall that the latter is the center of a circumcircle of a triangle spanned by three sites p1 , p2 , p3 . The coordinates of v can be described in terms of the coordinates of the pi , v = (Sx /a, Sy /a), with   kp1 k2 y1 1 1 Sx = det  kp2 k2 y2 1  2 kp3 k2 y3 1   x1 kp1 k2 1 1 det  x2 kp2 k2 1  Sy = 2 x3 kp3 k2 1   x1 y1 1 a = det  x2 y2 1  x3 y3 1

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It follows that the comparison between q = (xq , yq ) and v as above is determined by the expression sign(xq · a − Sx ) which is of degree 3. We can reduce the degree to 1 by the following observation: because we assume the query points to be of some bounded precision, say p, they all lie on some grid in R2 (with minimal distance 2−p between two grid points. Consider the grid with respect to precision p + 1, it contains the former grid and (among other points) a grid point in the center of every square spanned by the former grid (see Figure 9. Every Voronoi vertex is then “snapped” to the closest center point. Observe that all comparisons with query points yield the same result for the exact and the snapped Voronoi vertex. The x-coordinate of a snapped Voronoi vertex, however, can be represented by p + 1 bits, so that the degree of comparing the x-coordinates drops to 1.

Abbildung 9. The original grid is drawn by circular dots. The refined grid by square dots, where the possible positions of snapped Voronoi vertices are given by filled squares. Voronoi vertices, marked by an x, are snapped to the closest filled square as denoted by the dotted lines. The second predicate is the comparison of a query point to a Voronoi edge. Since a Voronoi edge connects two Voronoi vertices, we can express this test as an orientation test involving the query point and two Voronoi vertices. Using the exact representation v = (Sx /a, Sy /a) as above, this yields a polynomial of degree 6. Note that the snapping idea does not work here, because a query point above the original edge might be below the snapped edge. Still, the predicate can be computed with degree 2 instead: for that,

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remember the definition of a Voronoi edge: it is a bisector of two sites, say p1 and p2 (Figure 10). Assume that p1 lies above p2 (that is, has larger ycoordinate). Then, q is above the Voronoi edge if it is closer to p2 . Hence, the predicate can be computed by simply calling is_closer_to(q, p1 , p2 ) which is of degree 2. q

p1

p2

Abbildung 10. Instead of comparing q with the drawn Voronoi edge directly (involving the two Voronoi vertices drawn as square dots), q can be compared to the corresponding sites p1 and p2 . To summarize, using a slab decomposition, a closest site query can be answered with O(log n) time and degree 2. However, the presented method uses quadratic space which is prohibitive in practice. Still, the same arguments can be used for more clever search structures (such as a trapezoidal decomposition, for instance) to answer such queries in same time and degree, using only O(n) space. The analysis shows that we need a working precision of roughly twice the input precision to answer all queries correctly. However, the preprocessing step, the construction of the Voronoi diagram, is still of degree 4, because of the use of the in-circle predicate. Recently, Millman and Snoeyink [4] have been shown that a slightly relaxed version of the Voronoi diagram can be computed in degree 2 in almost the same running time, and this relaxed version is sufficient to answer closest site queries.

3. Controlled Perturbation We present an alternative solution to the robustness problem. It is particularly applicable in the common scenario that the coordinates of input points come from inexact measurements or computations, so that there

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is some freedom to perturb them without loosing too much information. The idea is then to perturb the input such that the “bad” situations presented in Lecture 1 do not appear anymore. Problem statement We can phrase the setup in two similar versions, depending on which parameters are fixed and which are subject to optimization. In the first version, we fix a working precision and try to minimize the amount of perturbation in the following sense: Given input points p1 , . . . , pn , and algorithm A (accepting p1 , . . . , pn as input) and a working precision P , find δ > 0 and p˜1 , . . . , p˜n with kpi − p˜i k ≤ δ such that A applied on p˜1 , . . . , p˜n yields the correct result for the perturbed point set when evaluating all predicates with precision P. Informally, we try to find an instance close to the original one such that floating-point arithmetic does not lead to any mistakes. However, what “close” means depends on the δ that is returned. Alternatively, one can fix a maximal perturbation and looks for a sufficient precision: Given input points p1 , . . . , pn , and algorithm A and δ > 0, find P , p˜1 , . . . , p˜n with kpi − p˜i k ≤ δ such that A applied on p˜1 , . . . , p˜n yields the correct result for the perturbed point set when evaluating all predicates with precision P . The key tool for guaranteeing correctness is the concept of guarded predicates. Such a geometric predicate only returns a +1, −1, 0, if it can certify that the value is correct. Otherwise, it returns a different value ¿with the semantic that the computed result could not be certified. In the latter case, we also say that the guard fires. We present two ways of realizing guarded predicates which only take a small overhead compared to the naive evaluation with fixed-precision arithmetic. Forward error analysis 1 Remember that geometric predicates are computed as signs of polynomials evaluated in the coordinates of the input points. We derive an explicit bound on the numerical error caused by fixing the precision. Let f ∈ Z[x1 , . . . , xn ] be some polynomial and let E := f (a1 , . . . , an ) be the exact result of evaluating f with some ai ∈ R. Fix a precision P . Let +fl , −fl , ·fl denote the addition, subtraction, and 1 This

paragraph is a summary of [5]

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˜ E ˜sup multiplication operations performed with precision P . We define E, and indE by the following recursive scheme: ˜ ˜sup E E E indE ai ai |ai | 0 ˜ A˜sup +fl B ˜sup 1 + max(indA , indB ) A + B A˜ +fl B ˜ A˜sup +fl B ˜sup 1 + max(indA , indB ) A − B A˜ −fl B ˜ ˜ ˜ ˜ A·B A ·fl B Asup ·fl Bsup 1 + indA + indB ˜ is the result of evaluating f (a1 , . . . , an ) in precision P . Note that E ˜sup and indE yield an upper bound on the difference between E and E ˜ E by ˜ |E − E|

≤

˜sup · indE B := 2−P · E

(1)

The proof works by (low-level) numerical analysis arguments and induction on the expression tree for E. We refer to the paper by Burnikel et al. for details. The bound can be extended to more complicated expressions involving divisions, square roots and k-th root operations, but we do not need it for our purposes. An immediate consequence of (1) is formulated next ˜ > B, then E > 0. If E ˜ < −B, then E < 0. Proposition. If E As an example, recall the orientation predicate from Lecture 1. We have E := (qx − px )(ry − py ) − (qy − py )(rx − px ) Fix p = (1, 1), q = (2, −1), and r = (4, 3) and a precision of 52 bits. In ˜ evaluates to 8. Moreover, E ˜sup and indE evaluate to 22 this example, E and 4, as demonstrated in Figure 11. Hence, we have B = 2−52 · 4 · 22 = ˜ > B, so that we can guarantee that E > 0 as 11 · 2−49 . It follows that E well. (Of course, no rounding error happens in this very simple example, ˜ = E). so that E Based on the proposition above, we can formulate a general guarded predicate as follows: Given f and a1 , . . . , an and P , the goal is to evaluate ˜ E ˜sup and indE as the sign of E := f (a1 , . . . , an ). We first evaluate E, −P ˜ ˜ > B, we return above with precision P , and set B := 2 Esup indE . If E ˜ ˜ +1. If E < −B, we return −1. If −B ≤ E ≤ B, we return ¿, denoting that the sign of E cannot be certified. The overhead compared to the unguarded version (which simply re˜ is roughly a factor of 2 in practice. turns the sign of E)

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indE

3 1 1 0 0 0 0 ( qx − px ) ( ry − py ) 2 1 3 1 4 3

3

−

10

12 E˜sup

1 1 0 0 0 0 ( qy − py ) ( rx − px ) 4 1 1 1 2 5

22

˜sup and indE . Abbildung 11. Example for the computation of E Interval arithmetic A different way to design certified primitives is to incorporate the error bounds in the computation using so-called interval arithmetic. For some a ∈ R and precision P , we define ← − a → − a

:=

max{b | b ≤ a, b is of precision P }

:=

min{b | b ≥ a, b is of precision P }

For intervals [a, b], [c, d] in R with a, b, c, d of precision p, define ←−−− −−−→ [a + c, b + d] ←−−− −−→ [a, b] [c, d] := [a − d, b − c] ←−−−−−−−−−−−−− −−−−−−−−−−−−−→ [a, b] ⊗ [c, d] := [min{ac, ad, bc, bd}, max{ac, ad, bc, bd}] [a, b] ⊕ [c, d]

:=

˜ be the For an expression E consisting of +, −, ·, and real values, let E expression defined by replacing + by ⊕, − by , · by ⊗, and a ∈ R by − − [← a ,→ a ]. The operations are designed to satisfy the following property, which can easily be proven by induction on the expression tree. Proposition. Let e ∈ R be the value of E and [emin , emax ] be the ˜ Then e ∈ [emin , emax ]. value of E. As an example, we consider an example where P = 4 (that is, we allow mantissa of length 3, plus one sign bit only), and E = (12 − 30) ·

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(17 + 14) − 51. Note that 12, 30, and 14 can be represented by 4 bit mantissa length (e.g., 30 = (1.111)2 · 23 ), but 17 and 51 cannot. For ← − instance, 51 = (1.10011)2 · 25 , so 51 = (1.100)2 · 25 = 32 + 16 = 48 and → − 51 = (1.101)2 · 25 = 32 + 16 + 4 = 52. We get ˜ E

=

([12, 12] [30, 30]) ⊗ ([16, 18] ⊕ [14, 14]) [48, 52]

([−18, −18] ⊗ [30, 32]) [48, 52] ←−−− −−−→ = ([−576, −540] [48, 52] =

([−576, −512] [48, 52] ←−−− −−−→ = ([−628, −560] =

=

([−640, −512]

The exact result is (−18 · 31) − 51 = −609. Interval arithmetic allows the simple design of a guarded predicate: with input polynomial f , coordinates a1 , . . . , an and precision P as usu˜ with the just described method. If the al, we evaluate the interval E resulting interval does not contain zero, all values in the interval have the same sign. In particular, the exact value has this sign, so we return ˜ we return ¿. the sign as a certified answer. If 0 ∈ E, Interval arithmetic is an attractive variant because of its simplicity and because the overhead is bounded by roughly a factor of 4. However, the intervals tend to get pretty large for expression of large degree. This causes the frequent appearance of zero in the resulting interval and reduces the effectiveness of the approach. However, it can be shown that ˜ converges to zero. for P → ∞ and a fixed expression E, the width of E The controlled perturbation scheme We formulate a solution to our original setup. Let A be an algorithm with input a1 , . . . , an and running with a fixed precision P . We want to compute δ and perturbations a˜1 , . . . , a˜n with ka˜i − ai k < δ such that A applied on a˜1 , . . . , a˜n yields the correct result. We assume that all geometric predicate used in A are guarded with one of the methods presented before. Then, the algorithm proceed in the following steps: 1) Initialize δ ← 2−P 2) Choose a˜1 , . . . , a˜n at random with precision P and kai − a˜i k < δ.

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3) Evaluate A on input a˜1 , . . . , a˜n using guarded predicates. 4) If any predicate returns ¿ during the execution, set δ ← 2δ and goto step 2 5) Return δ, a˜1 , . . . , a˜n and the result of A on that input. The variant where δ is fixed and the precision is sought for is similar: start with P ← 4 and double the precision after every failed instance until no predicates returns ¿. It can be shown that the above algorithm terminates, because for δ large enough, the probability of termination is more than 12 . The main theoretical challenge is to quantify what “large enough” means in concrete instances of the problem. This has been analyzed for Delaunay triangulations [6] and for arrangements of circles [7]. A general framework to analyze a large class of algorithm has been presented in [8]. It should be finally mentioned that controlled perturbation can be interpreted as a method to avoid degenerate or nearly-degenerate instances, and is therefore not an adequate solution if a solution for the exact input is required (compare the section on “non-responses” in Lecture 1).

4. Exact Geometric Computation We discuss the most rigorous, and conceptually most straight-forward approach to address the robustness problem: exact computation. Perhaps surprisingly, rigorous exactness can be achieved with just a slight overhead compared to unreliable floating-point arithmetic. The EGC paradigm The control flow of a geometric algorithm depends on the evaluation of geometric predicates (like the orientation test in the convex hull algorithm of Lecture 1). The robustness problem can be reformulated by the problem that theoretical description of an algorithm and its implementation might deviate in their control flow. The Exact Geometric Computation (EGC) paradigm simply disallows that possibility: every geometric primitive should return the mathematically exact result in all cases (including degeneracies). The advantage of this approach is that the correctness of the theoretical algorithm directly carries over to the implementation without further analysis of the algorithm

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(like in Lecture 2), perturbing the input (like in Lecture 3), or changing the algorithm to ensure consistency. Recall the geometric primitive divide into two classes: predicates (for instance, the orientation test) and constructions (for instance, construction a Voronoi vertex). A weaker form of the EGC paradigm says that only predicates must be computed exactly whereas constructions are still allowed to be approximate. It should not be too surprising that this weak form is not robust in all cases because the algorithm could exploit implicit assumptions on the constructed object which are not satisfied by the approximate version. Filtered predicates The most obvious solution to satisfy the EGC paradigm is the usage of exact number types that do not suffer from rounding errors. For instance, the mpfr2 library provides a floating point type with arbitrary and dynamic mantissa length. This type can be used if all input points as well as all constructed points during the algorithm are double values, which is satisfied for the weak form of the EGC paradigm. Even more general, the gmp library3 provides an exact number type for arbitrary rational numbers (as a numerator-denominator pair of arbitrary-sized integer values). If all points in the computation stay rational, using this number types solves the robustness problem. The price to pay using the mentioned number types is a severe slowdown because every primitive computation takes more time. The EGC paradigm states, however, that every predicate (and every construction, in the strong form), must return the exact result. That does not mean, however, that all computations within the predicate must be performed exactly. In fact, one is allowed to use any technique that works fast, as long as the outcome is correct. We call the implementation of a predicate a filter if it returns either a certified answer or a value ¿ denoting that no certified answer could be given. The latter is called filter failure. Note that we have learned about filters in Lecture 3, where we called them “guarded predicates”. In the EGC context, we first try to evaluate the predicate using the filter and only evaluate exactly if a filter failure occurs. The challenge is to design filters that are efficiently computable but do not fail to often. Several 2 http://www.mpfr.org/ 3 http://gmplib.org/

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approaches for efficient realizations have been presented [9, 10], but they all eventually boil down to the techniques presented in Lecture 3. The cgal library The EGC paradigm is realized in the state-ofthe-art software package for computational geometry, the Computational Geometry Algorithms Library (cgal)4 . It provides functionality for most basic geometric data structures, including convex hull, Delaunay triangulations, Arrangements, Boolean set operations, and more. cgal separates the combinatorial layer of an algorithm from the numerical layer (the set of geometric primitives). The latter is encapsulated in a kernel which is a collection of primitives required by geometric algorithms (including all primitives discussed in this course). Geometric structures depend on a kernel which provides the primitives to construct the structure. One can visualize this mechanism by imagining that the construction method for the structure gets the kernel as additional input. For people that know C++, this is realized by making the data structure a template, instantiated by the kernel. The are various kernels provided in cgal: for instance the kernel Cartesian represents points by double coordinates and provides all primitives with unreliable double arithmetic. Replacing double by an exact type, like gmpq yields an exact (but slow) kernel evaluating all predicates with exact computations. Filters based on interval arithmetic can be enabled using the Filtered_kernel, which takes another kernel as an argument and uses that kernel in case of filter failures. For the convenience of the user, there are two default kernels: the first one, called Exact_predicate_exact_constructions internally uses an exact number type for rationals, but tries to make maximal use of filters during the computation. Alternatively, there is also Exact_predicate_inexact_constructions which realizes the weak form of the EGC paradigm. As an example, we consider a demo program to compute a Delaunay triangulation in Figure 12. Note that the Delaunay triangulation is instantiated with the Exact_predicate_exact_constructions kernel in 4 http://www.cgal.org

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line 7. The resulting algorithm is therefore certified. Exchanging line 6 to typedef CGAL::Cartesian K; yields a faster, but unreliable implementation using double arithmetic.5

Non-linear geometry The kernels mentioned so far are examples of linear kernel; all intermediate results can be represented using rational coordinates. However, one leaves this domain immediately when considering anything more complicated than points and segments: the intersection point of a circle (with rational center and radius) and a line has coordinates that involve square roots. We can know imagine constructing three points of this form, define the unique circle going these points and intersecting it with another circle of this type. How would we represent the coordinates of the intersections? A number α ∈ R is called (real) algebraic if there exists a f ∈ Z[x] with f (α) = 0. If f is irreducible with this property, it is called a minimal polynomial for α. The √ degree of α is the degree of the minimal polynomial. As an example, 2 is algebraic of degree 2 because it is a root of x2 − 2. Moreover, every rational pq is algebraic of degree 1 because qx − p is a minimal polynomial. Prominent examples of non-algebraic numbers are π and the Euler number e. Note that the coordinates from the construction above can be expressed by equations in the original input points (which are supposed to be rational), and are therefore algebraic. √ General algebraic numbers can be represented exactly; from the 2, it might appear attractive to represent algebraic numbers as square-root expressions. While this is possible for numbers up to degree 4, it has been known for almost 200 years that algebraic number of higher degree cannot be represented by such an expression. A common possibility is described next. Let α be algebraic and f ∈ Z[x] with f (α) = 0 (f is not assumed to be minimal). We call an interval I isolating for α and f if α ∈ I, and no other root of f lies in I. Note that a polynomial of degree d has at most d roots, hence an isolating interval for α and f always exists. In turn, an isolating interval I and a polynomial f determine an algebraic number uniquely. We take the pair (f, I) as a representation 5 Line

1 needs also be adapted to “#include ”

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#include #include #include typedef CGAL::Exact_predicates_exact_constructions_kernel K; typedef CGAL::Delaunay_triangulation_2 Delaunay; typedef K::Point_2

Point;

int main() { std::ifstream in("example.txt"); std::istream_iterator begin(in); std::istream_iterator end; Delaunay dt; dt.insert(begin, end); std::cout 0. i,j

Pick n vectors y1 , . . . , yn in R2 with coordinates yi = (cos mϕi , sin mϕi ). If y = u1 y1 + . . . + un yn , then X < y, y > = ||y||2 = ui uj cos mϕi,j > 0. i,j

This inequality and the inequalities cm,k > 0 complete our proof. Proof of Lemma 2. Proof. The expansion of f in terms of Pk is f=

9 X k=0

ck Pk = P0 + 1.6P1 + 3.48P2 + 1.65P3 + 1.96P4 + 0.1P5 + 0.32P9 .

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We have c0 = 1, ck > 0, k = 1, 2, . . . , 9. Using Lemma 1 we get S(X) =

9 X k=0

ck

n n X X

Pk (cos(φi,j )) >

n n X X

c0 P0 = n2 .

i=1 j=1

i=1 j=1

Proof of Lemma 3. Proof. 1. The polynomial f (t) satisfies the following properties (see Fig.2): (i) f (t) is a monotone decreasing function on the interval [−1, −t0 ]; (ii) f (t) < 0 for t ∈ (−t0 , 1/2]; where f (−t0 ) = 0, t0 ≈ 0.5907. These properties hold because f (t) has the only one root −t0 on [−1, 1/2], and there are no zeros of the derivative f 0 (t) (8th degree polynomial) on [−1, −t0 ].

Let Si (X) :=

n X

f (cos(φi,j )), then S(X) =

j=1

n X

Si (X). From this follows

i=1

if Si (X) < 13 for i = 1, 2, . . . , n, then S(X) < 13n. We obviously have φi,i = 0, so f (cos φi,i ) = f (1). Note that our assumption on X (φi,j > 60◦ , i 6= j) yields cos φi,j 6 1/2. Therefore, cos φi,j lies in the interval [-1,1/2]. Since (ii), if cos φi,j ∈ [−t0 , 1/2], then f (cos φi,j ) 6 0. Let J(i) := {j : cos φi,j ∈ [−1, −t0 )}. We obtain X Si (X) 6 Ti (X) := f (1) + f (cos φi,j ). (1) j∈J(i)

Let θ0 = arccos t0 , θ0 ≈ 53.794◦ . Then j ∈ J(i) iff φi,j > 180◦ − θ0 , i.e. θj < θ0 , where θj = 180◦ − φi,j . In other words all xi,j , j ∈ J(i) lie inside the circle of center e0 and radius θ0 , where e0 = −xi is the antipodal point to xi . 2. Let us consider on S2 points e0 , y1 , . . . , ym such that φi,j = dist(yi , yj ) > 60◦ for all i 6= j,

dist(e0 , yi ) 6 θ0 for 1 6 i 6 m. (2)

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Denote by µ the highest value of m such that the constraints in (2) define a non-empty set of points y1 , . . . , ym . Suppose 0 6 m 6 µ and Y = {y1 , . . . , ym } satisfies (2). Let H(Y ) = H(y1 , . . . , ym ) := f (1) + f (− cos θ1 ) + . . . + f (− cos θm ), θi = dist(e0 , yi ) hm := max H(Y ), Y

hmax := max {h0 , h1 , . . . , hµ }. It is clear that Ti (X) 6 hm , where m = |J(i)|. From (1) it follows that Si (X) 6 hm . Thus, if we prove that hmax < 13, then we prove Lemma 3. 3. Now we prove that µ 6 4. Suppose Y = {y1 , . . . , ym } ⊂ S2 satisfies (2). If e0 is the North pole and yi has polar coordinates (θi , ϕi ), then from the law of cosines we have: cos φi,j = cos θi cos θj + sin θi sin θj cos(ϕi − ϕj ). From (2) we have cos φi,j 6 1/2, then cos(ϕi − ϕj ) 6 Let

Q(α) =

1/2 − cos θi cos θj . sin θi sin θj

1/2 − cos α cos β , sin α sin β

then Q0 (α) =

(3) 2 cos β − cos α . 2 sin2 α sin β

From this follows, if 0 < α, β 6 θ0 , then cos β > 1/2 (because θ0 < 60◦ ); so then Q0 (α) > 0, and Q(α) 6 Q(θ0 ). Therefore, 1/2 − cos θi cos θj 1/2 − cos2 θ0 1/2 − t20 = . 6 2 sin θi sin θj 1 − t20 sin θ0 Combining this inequality and (3), we get cos(ϕi − ϕj ) 6

1/2 − t20 . 1 − t20

Note that arccos((1/2 − t20 )/(1 − t20 )) ≈ 76.582◦ > 72◦ . Then m 6 4 because no more than four points can lie in an unit circle with the minimum angular separation between any two points greater than 72◦ .

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4. Now we have to prove that hmax = max {h0 , h1 , h2 , h3 , h4 } < 13. We obviously have h0 = f (1) = 10.11 < 13. From (i) follows that f (− cos θ) is a monotone decreasing function in θ on [0, θ0 ]. Then for m = 1 : H(y1 ) = f (1) + f (− cos θ1 ) attains its maximum at θ1 = 0, h1 = f (1) + f (−1) = 12.88 < 13 5. Let us consider for m = 2, 3, 4 an optimal arrangement {e0 , y1 , . . . , ym } in S2 that gives maximum of H(Y ) = hm . Note that for optimal arrangement points yk cannot be shifted towards e0 because in this case H(Y ) increases. For m = 2 this yields: e0 ∈ y1 y2 , and dist(y1 , y2 ) = 60◦ . If e0 ∈ / y1 y2 , then whole arc y1 y2 can be shifted to e0 . Also if dist(y1 , y2 ) > 60◦ , then y1 (and y2 ) can be shifted to e0 . For m = 3 we prove that ∆3 = y1 y2 y3 is a spherical regular triangle with edge length 60◦ . As above, e0 ∈ ∆3 , otherwise whole triangle can be shifted to e0 . Suppose dist(y1 , yi ) > 60◦ , i = 2, 3, then dist(y1 , e0 ) can be decreased. From this follows that for any yi at least one of the distances {dist(yi , yj )} is equal to 60◦ . Therefore, at least two sides of ∆3 (say y1 y2 and y1 y3 ) have length 60◦ . Also dist(y2 , y3 ) = 60◦ , conversely y3 (or y2 , if e0 ∈ y1 y3 ) can be rotated about y1 by a small angle towards e0 (Fig.3). When m = 4 first we prove that ∆4 = y1 y2 y3 y4 is a convex quadrangle. Conversely, we may assume that y4 ∈ y1 y2 y3 . The great circle that is orthogonal to the arc e0 y4 divides S2 into two hemispheres: H1 and H2 . Suppose e0 ∈ H1 , then at least one yi (say y3 ) belongs H2 (Fig.4). So the angle ∠e0 y4 y3 greater than 90◦ , then (again from the law of cosines) dist(y3 , e0 ) > dist(y3 , y4 ). Thus, θ3 = dist(y3 , e0 ) > dist(y3 , y4 ) > 60◦ > θ0 − a contradiction. Arguing as for m = 3 it is easy to prove that ∆4 is a spherical equilateral quadrangle (rhomb) with edge length 60◦ .

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 s s y@ 3 e0 @ 60◦@ s y@ 1

s y2 60◦

sy3 H2 aasy4 a H1 se0 aa asa s y1 y2

aa

Fig. 3

173

ppsy3 ppp

J p

 ppp J ◦  ppp 60 θ pp J 60◦

3  psyc J

θ1 ` s `θ2 J y1

``` Jsy2  s  e0

Fig. 4

Fig. 5

6. Now we introduce the function F1 (ψ),2 where ψ ∈ [60◦ , 2θ0 ]: F1 (ψ) :=

max {Fe1 (θ, ψ)},

ψ/26θ6θ0

Fe1 (θ, ψ) = f (− cos θ) + f (− cos(ψ − θ)).

So if dist(yi , yj ) = ψ, then f (− cos θi ) + f (− cos θj ) 6 F1 (ψ).

(4)

Therefore, H(y1 , y2 ) 6 h2 = f (1) + F1 (60◦ ) ≈ 12.8749 < 13. 7. When m = 4, ∆4 is a spherical rhomb. Let d1 = dist(y1 , y3 ), and d2 = dist(y2 , y4 ), then cos(d1 /2) cos(d2 /2) = 1/2 (Pythagorean theorem, the diagonals of ∆4 are orthogonal). So if ρ(s) := 2 arccos[1/(2 cos(s/2))], then d1 = ρ(d2 ), d2 = ρ(d1 ), ρ(90◦ ) = 90◦ . Suppose d1 6 d2 . Since θi 6 θ0 , d2 6 2θ0 , then ρ(2θ0 ) 6 d1 6 90◦ 6 d2 6 2θ0 . Now we consider two cases: 1) ρ(2θ0 ) 6 d1 < 77◦ , and 2) 77◦ 6 d1 6 90◦ . 1) F1 (ψ) is a monotone decreasing function in ψ. Then (4) implies f (− cos θ1 ) + f (− cos θ3 ) 6 F1 (ρ(2θ0 )), 2 For given ψ, the value F (ψ) can be find as the maximum of the 9th degree 1 polynomial Ω(s) = Fe1 (θ, ψ), s = cos (θ − ψ/2), on the interval [cos(θ0 − ψ/2), 1].

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f (− cos θ2 ) + f (− cos θ4 ) < F1 (ρ(77◦ )), so then H(Y ) < f (1) + F1 (ρ(2θ0 )) + F1 (ρ(77◦ )) ≈ 12.9171 < 13. 2) In this case we have H(Y ) 6 f (1) + F1 (77◦ ) + F1 (90◦ ) ≈ 12.9182 < 13. Thus, h4 < 13. 8. Our last step is to show that h3 < 13.3 Since ∆3 is a regular triangle, H(Y ) = f (1)+f (− cos θ1 )+f (− cos θ2 )+ f (− cos θ3 ) is a symmetric function in θi , so we can consider only the case θ1 6 θ2 6 θ3 6 θ0 . p In this case R0 6 θ3 6 θ0 , where R0 = arccos 2/3 ≈ 35.2644◦ . (Note that the circumradius of ∆3 equals R0 .) Let yc is the center of ∆3 . Denote by u the angle ∠e0 y3 yc . Then (see Fig.5) cos θ1 = cos 60◦ cos θ3 + sin 60◦ sin θ3 cos (R0 − u), cos θ2 = cos 60◦ cos θ3 + sin 60◦ sin θ3 cos (R0 + u),

√ where ∠y1 y3 yc = ∠y2 y3 yc = R0 , 0 6 u 6 u0 = arccos(cot θ3 / 3) − R0 (if u = u0 , then θ2 = θ3 ). For fixed θ3 = ψ, H(y1 , y2 ) becomes the polynomial of degree 9 in s = cos u. Denote by F2 (ψ) the maximum of this polynomial on the interval [cos u0 , 1]. Let {ψ1 , . . . , ψ6 } = {R0 , 38◦ , 41◦ , 44◦ , 48◦ , θ0 }. It’s clear that F2 (ψ) is a monotone increasing function in ψ on [R0 , θ0 ]. From other side, f (− cos ψ) is a monotone decreasing function in ψ. Therefore for θ3 ∈ [ψi , ψi+1 ] we have H(Y ) = H(y1 , y2 ) + f (− cos θ3 ) < wi := F2 (ψi+1 ) + f (− cos ψi ). 3 More

detailed analysis shows h3 ≈ 12.8721, h4 ≈ 12.4849.

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Since, {w1 , . . . , w5 } ≈ {12.9425, 12.9648, 12.9508, 12.9606, 12.9519}, we get h3 < max{wi } < 13. Thus, hm < 13 for all m as required.

3

Delsarte’s method

Let X = {x1 , x2 , . . . , xM } be any finite subset of the unit sphere Sn−1 ⊂ Rn , Sn−1 = {x : x ∈ Rn , x · x = ||x||2 = 1}. From here on we will speak of x ∈ Sn−1 alternatively of points in Sn−1 or of vectors in Rn . By φij we denote the spherical (angular) distance between xi , xj . It is clear that for any real numbers u1 , u2 , . . . , uM the relation X X || ui xi ||2 = cos φij ui uj ≥ 0 i,j

holds, or equivalently the Gram matrix T (X) is positive semidefinite, where T (X) = (tij ), tij = cos φij = xi · xj . Schoenberg [31] extended this property to Gegenbauer (ultraspheri(n) (n) cal) polynomials Gk of tij . He proved that if gij = Gk (tij ), then the matrix (gij ) is positive semidefinite. Schoenberg proved also that the converse holds: if f (t) is a real polynomial and for any finite X ⊂ Sn−1 (n) the matrix (f (tij )) is positive semidefinite, then f is a sum of Gk with nonnegative coefficients. Let us recall the definition of Gegenbauer polynomials. Suppose (n) Ck (t) be the polynomials defined by the expansion (1 − 2rt + r2 )1−n/2 =

∞ X

(n)

rk Ck (t).

k=0 (n)

(n)

(n)

Then the polynomials Gk (t) = Ck (t)/Ck (1) are called Gegenbauer (n) or ultraspherical polynomials. (So the normalization of Gk is deter(n) mined by the condition Gk (1) = 1.)

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Also the Gegenbauer polynomials Gk formula:

can be defined by recurrence (n)

(n) G0

(n) G1

= 1,

= t, . . . ,

(n) Gk

(n)

(2k + n − 4) t Gk−1 − (k − 1) Gk−2 = k+n−3

They are orthogonal on the interval [−1, 1] with respect to the weight function ρ(t) = (1 − t2 )(n−3)/2 (see details in [9, 11, 17, 31]). In the case (n) (4) n = 3, Gk are Legendre polynomials Pk , and Gk are Chebyshev polynomials of the second kind (but with a different normalization than usual, Uk (1) = 1), (4)

Gk (t) = Uk (t) =

sin ((k + 1)φ) , (k + 1) sin φ

t = cos φ,

k = 0, 1, 2, . . .

For instance, U0 = 1, U1 = t, U2 = (4t2 − 1)/3, U3 = 2t3 − t, U4 = (16t4 − 12t2 + 1)/5, . . . , U9 = (256t9 − 512t7 + 336t5 − 80t3 + 5t)/5. Let us now prove the bound of Delsarte’s method. If a P matrix (gij ) is positive semidefinite, then for any realP ui the inequality gij ui uj > 0 holds, and then for ui = 1, we have gij ≥ 0. Therefore, for gij = i,j (n) Gk (tij ),

we obtain M X M X

(n)

Gk (tij ) > 0

(3.1)

i=1 j=1

Suppose (n)

(n)

f (t) = c0 G0 (t) + . . . + cd Gd (t), where c0 > 0, . . . , cd > 0. PP Let S(X) = f (tij ). Using (3.1), we get i

S(X) =

(3.2)

j

d X M X M X k=0 i=1 j=1

(n)

ck Gk (tij ) >

M X M X

(n)

c0 G0 (tij ) = c0 M 2 .

(3.3)

i=1 j=1

Let X = {x0 , . . . , xM } ⊂ Sn−1 be a spherical z-code, i.e. for all i 6= j, tij = cos φij = xi · xj 6 z, i.e. tij ∈ [−1, z] (but tii = 1). Suppose f (t) 6 0 for t ∈ [−1, z], then S(X) = M f (1) + 2f (t12 ) + . . . +

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2f (tM −1 M ) 6 M f (1). If we combine this with (3.2), then for c0 > 0 we get f (1) (3.4) M6 c0 The inequality (3.4) play a crucial role in the Delsarte method (see details in [3, 4, 6, 11, 15, 16, 23, 25, 29]). If z = 1/2 and c0 = 1, then (3.4) implies k(n) 6 f (1). In [25, 29] Levenshtein, Odlyzko and Sloane have found the polynomials f (t) such that f (1) = 240, when n = 8; and f (1) = 196 560, when n = 24. Then k(8) 6 240, k(24) 6 196 560. When n = 8, 24, there exist sphere packings (E8 and Leech lattices) with these kissing numbers. Thus k(8) = 240 and k(24) = 196 560. When n = 4, a polynomial f of degree 9 with f (1) = 25.5585... was found in [29]. This implies 24 6 k(4) 6 25.

4

An extension of Delsarte’s method.

Let us now generalize the Delsarte bound M 6 f (1)/c0 . Definition. Let f (t) be any function on the interval [−1, 1]. Consider on Sn−1 points y0 , y1 , . . . , ym such that yi · yj 6 z for all i 6= j,

f (y0 · yi ) > 0 for 1 6 i 6 m.

(4.1)

Denote by µ = µ(n, z, f ) the highest value of m such that the constraints in (4.1) define a non-empty set of points (y0 , . . . , ym ). Suppose 0 6 m 6 µ. Let H(Y ) = H(y0 ; y1 , . . . , ym ) := f (1) + f (y0 · y1 ) + . . . + f (y0 · ym ), hm := max{H(Y )}, Y

hmax := max {h0 , h1 , . . . , hµ }.

Remark. hmax depends on n, z, and f. Throughout this paper it is clear what f, n, and z are; so we denote by hmax the value hmax (n, z, f ). Theorem 2. Suppose X ⊂ Sn−1 is a spherical z-code, |X| = M, and (n) (n) f (t) = c0 G0 (t) + . . . + cd Gd (t), where c0 > 0, c1 > 0, . . . , cd > 0. Then 1 hmax = max{h0 , h1 , . . . , hµ }. M6 c0 c0

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Proof. Since f satisfies (3.2), then (3.3) yields S(X) > c0 M 2 . Let J(i) := {j : f (xi · xj ) > 0, j 6= i}, and X(i) = {xj : j ∈ J(i)}. Then Si (X) =

M X

X

f (xi · xj ) 6 f (1) +

j=1

f (xi · xj ) = H(xi ; X(i)) 6 hmax ,

j∈J(i)

so then S(X) =

M X

Si (X) 6 M hmax .

i=1

We have c0 M 2 6 S(X) 6 M hmax , i.e. c0 M 6 hmax as required. Note that h0 = f (1). If f (t) 6 0 for all t ∈ [−1, z], then for a z-code X we have µ = 0, i.e. hmax = h0 = f (1). Therefore, this theorem yields the Delsarte bound M 6 f (1)/c0 . The problem of evaluating of hmax in general case looks even more complicated than the upper bound problem for spherical z-codes. It is not clear how to find µ? Here we consider this problem only for a very restrictive class of functions f (t): f (t) 6 0 for t ∈ [−t0 , z], t0 > z > 0. Let us denote by A(k, ω) the maximal number of points in a spherical s-code Ω ⊂ Sk−1 of minimal angle ω, cos ω = s. (Note that A(n, 60◦ ) is the kissing number k(n).) Theorem 3. Suppose Y = {y1 , . . . , ym } is a spherical z-code in Sn−1 , and points yi lie inside the sphere of center e0 and radius θ0 , where t0 = cos θ0 > z. Then  z − t20  m 6 A n − 1, arccos . 1 − t20 Proof. We have φi,j = dist(yi , yj ) > δ = arccos z for i 6= j; θi = arccos(e0 · yi ) 6 θ0 for 1 6 i 6 m; and θ0 6 δ.

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Let Π be the projection of Y onto equator Sn−2 from pole e0 . Denote by γi,j the distances between points of Π in Sn−2 . Then from the law of cosines and the inequality cos φi,j 6 z, we get cos γi,j = Let

Q(α) =

cos φi,j − cos θi cos θj z − cos θi cos θj 6 sin θi sin θj sin θi sin θj

z − cos α cos β , sin α sin β

cos β − z cos α . sin2 α sin β

then Q0 (α) =

From this follows, if 0 < α, β 6 θ0 , then cos β > z (because θ0 6 δ); so then Q0 (α) > 0, and Q(α) 6 Q(θ0 ). Therefore, cos γi,j 6

z − cos θi cos θj z − cos2 θ0 z − t20 6 = sin θi sin θj 1 − t20 sin2 θ0

that complete our proof. Corollary 1. Suppose f (t) 6 0 for t ∈ [−t0 , z],

t0 > z > 0, then

 z − t20  µ(n, z, f ) 6 A n − 1, arccos . 1 − t20 Proof. The assumption on f yields f (y0 · yi ) > 0 only if θi = dist(e0 , yi ) < θ0 = arccos t0 , where e0 = −y0 is the antipodal point to y0 . Therefore, this set of points {e0 , y1 , . . . , ym } satisfies the assumptions in Theorem 3. The next claim will be applied to prove that k(4) = 24. Corollary 2. Suppose f (t) 6 0 for t ∈ [−t0 , 1/2],

t0 > 0.6058, then

µ = µ(4, 1/2, f ) 6 6. Proof. Note that for t0 > 0.6058, arccos[(1/2 − t20 )/(1 − t20 )] > 77.87◦ . So Corollary 1 implies µ(4, 1/2, f ) 6 A(3, 77.87◦ ). Denote by ϕk (M ) the largest angular separation that can be attained in a spherical code on Sk−1 containing M points. In three dimensions the best codes and the values ϕ3 (M ) presently known for M 6 12 and

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M = 24 (see [13, 18, 32]). For instance, Sch¨ utte and van der Waerden [32] proved that ϕ3 (5) = ϕ3 (6) = 90◦ and ϕ3 (7) ≈ 77.86954◦ (cos ϕ3 (7) = cot 40◦ cot 80◦ ). Since 77.87◦ > ϕ3 (7), then A(3, 77.87◦ ) < 7, i.e. µ 6 6. Corollary 1 shows that if t0 is close enough to 1, then µ is small enough. Then one gets relatively small - dimensional optimization problems for computation of numbers hm for small n. If additionally f (t) is a monotone decreasing function on [−1, −t0 ], then these problems can be reduced to low-dimensional optimization problems of a type that can be treated numerically.

5

Optimal sets for monotonic functions

In this section we consider f (t) that satisfies the monotonicity assumption: f (t) is a monotone decreasing function on the interval [−1, −t0 ], f (t) 6 0 for t ∈ [−t0 , z],

t0 > z > 0

(∗)

. Consider on Sn−1 points y0 , y1 , . . . , ym that satisfy (4.1). Denote by θk for k > 0 the distance between yk and e0 , where e0 = −y0 is the antipodal point to y0 . Then y0 · yk = − cos θk , and H(Y ) is represented in the form: H(Y ) = f (1) + f (− cos θ1 ) + . . . + f (− cos θm ).

(5.1)

A subset C of Sn−1 is called (spherical) convex if it contains, with every two nonantipodal points, the small arc of the great circle containing them. If, in addition, C does not contain antipodal points, then C is called strongly convex. The closure of a convex set is convex and is the intersection of closed hemispheres (see details in [14]). If a subset Z of Sn−1 lies in a hemisphere, then the convex hull of Z is well defined, and is the intersection of all convex sets containing Z. Suppose f (t) satisfies (∗), then Qm = {y1 , . . . , ym } lies in the hemisphere of center e0 . Denote by ∆m the convex hull of Qm in Sn−1 , ∆m = conv Qm .

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Now we consider an optimal arrangement of Qm for H. Let δ = arccos z, ˜ (Qm ) = number of φi,j = δ (yi · yj = z). φi,j = dist(yi , yj ), N Definition We say that Qm is optimal if H(Y ) = hm . If optimal Qm is not unique up to isometry, then we call Qm as optimal if it has maximal ˜ (Qm ). N The function f (t) is monotone decreasing on [−1, −t0 ]. By (5.1) it follows that the function H(Y ) increases whenever θk decreases. This means that for an optimal Qm no yk ∈ Qm can be shifted towards e0 . That yields e 0 ∈ ∆m (5.2) because in the converse case whole Qm can be shifted to e0 . From this follows that for m = 1, e0 = y1 . Thus h1 = f (1) + f (−1). It was proved in Section 2 that for m = 2 : dist(y1 , y2 ) = δ, thus h2 = f (1) +

max {(f (− cos θ) + f (− cos(δ − θ))}, δ/26θ6θ0

θ0 = arccos t0 .

It was also proved that ∆3 is a spherical regular triangle with edge length δ. Using similar arguments it’s not hard to prove that for n > 3, ∆4 is a spherical regular tetrahedra with edge length δ. 4 Let ∆m , m 6 n, is a spherical regular simplex with edge length δ, and Ωm = {y : y ∈ ∆m , y · yk > t0 , 1 6 k 6 m}. Note that Ωm is a convex set in Sn−1 . Let Hm (y) = f (1) + f (−y · y1 ) + . . . + f (−y · ym ). Then hm is the maximum of Hm (y) on Ωm . hm = max Hm (y), y∈Λm

Ωm ⊂ ∆m ⊂ Sn−1 ,

2 6 m 6 min(n, µ).

(5.3)

4 For m 6 n, ∆ m is a spherical regular simplex with edge length δ. In this paper we need just cases m = 3, 4.

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When n > m any yk ∈ Qm is a vertex of ∆m . In other words, no yk that lies inside ∆m . In fact, that has been proved in Section 2 (see 5, Fig. 4). In the first version of the paper [26] has been claimed that for optimal Qm with m > n, for any yk ∈ Qm there are at least n − 1 distinct points in Qm at the distance of δ from yk . However, Eiichi Bannai and Makoto Tagami found some gaps in our exposition. Most of them are related to “degenerated” configurations. In this paper we need only the case n = 4, m = 5. For this case they verified each step of our proof, considered all “degenerated” configurations, and finally gave clean and detailed proof. I wish to thank Eiichi Bannai and Makoto Tagami for this work. Now this claim in general case can be considered only as conjecture.

6

An algorithm for computation suitable polynomials f (t)

In this section is presented an algorithm for computation “optimal” 5 polynomials f such that f (t) is a monotone decreasing function on the interval [−1, −t0 ], and f (t) 6 0 for t ∈ [−t0 , z], t0 > z > 0. This algorithm based on our knowledge about optimal arrangement of points yi for given m. Coefficients ck can be found via discretization and linear programming; such method had been employed already by Odlyzko and Sloane [29] for the same purpose. Let us have a polynomial f represented in the form f (t) = 1 + d P (n) ck Gk (t). We have the following constraints for f : (C1) ck >

k=1

0, 1 6 k 6 d; (C2) f (a) > f (b) for −1 6 a < b 6 −t0 ; (C3) f (t) 6 0 for −t0 6 t 6 z. When m 6 n, hm = max Hm (y), y ∈ Λm . We do not know y where Hm attains its maximum, so for evaluation of hm let us use yc − the center of ∆m . All vertices yk of ∆m are at the distance of Rm from yc , where 5 Open problem: is it true that for given t , d this algorithm defines f with minimal 0 hmax ?

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p cos Rm = (1 + (m − 1)z)/m. When m = 2n − 2, ∆m presumably is a regular (n − 1)-dimensional √ cross-polytope. (It is not S proven yet.) In this case cos Rm = z. Let In = {1, . . . , n} {2n − 2}, m ∈ In , bm = − cos Rm , whence Hm (yc ) = f (1) + mf (bm ). If F0 is such that Hm (y) 6 E = F0 + f (1), then (C4) f (bm ) 6 F0 /m, m ∈ In . A polynomial f that satisfies (C1-C4) and gives the minimal E (note that E = F0 + 1 + c1 + . . . + cd = F0 + f (1) will become a lower estimate of hmax ) can be found by the following Algorithm. Input: n, z, t0 , d, N. Output: c1 , . . . , cd , F0 , E. First replace (C2) and (C3) by a finite set of inequalities at the points aj = −1 + j, 0 6 j 6 N,  = (1 + z)/N : Second use linear programming to find F0 , c1 , . . . , cd so as to minimize d P E − 1 = F0 + ck subject to the constraints k=1

ck > 0, d X

(n)

ck Gk (aj ) >

k=1

d X

1 6 k 6 d; (n)

ck Gk (aj+1 ),

aj ∈ [−1, −t0 ];

k=1

1+

d X

(n)

aj ∈ [−t0 , z];

(n)

m ∈ In .

ck Gk (aj ) 6 0,

k=1

1+

d X

ck Gk (bm ) 6 F0 /m,

k=1

Let us note again that E = max Hm (yc ) 6 hmax here, and that m∈In

E = hmax only if hmax = Hm0 (yc ) for some m0 ∈ In .

7

On calculations of hm for m 6 n

Here we explain how to solve the optimization problem (5.3). Let ∆m ⊂ Sm−1 is a spherical regular simplex with edge length

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δ = arccos z; yi , i = 1, . . . , m, are the vertices of ∆m ; ti = y · yi = cos θi > t0 = cos θ0 ; t0 > z; f (t) is a monotone decreasing function on the interval [−1, −t0 ]; hm is the maximum of Hm (y) subject to the constraints ti > t0 ; Hm (y) = f (1) + f (−y · y1 ) + . . . + f (−y · ym ). The first method. Hm (y) is a symmetric function in the variables θ1 , . . . , θm . Then we can consider this problem only on the domain Λ = {y : θm 6 . . . 6 θ2 6 θ1 }. Note that Λ is a spherical simplex. Let us consider a barycentric triangulation of this simplex such that the diameter of any simplex σi of this triangulation is not exceed . It is easy to prove that for any yk , y ·yk attains its maximum on σi at some vertex of σi . Denote this vertex by yk,i . Let I = {i : y1,i · y1 > t0 }. So for i ∈ I we have f (−yk,i · yk ) = max {f (−y · yk )}, y∈σi

then hm 6 max i∈I

m nX

o f (−yk,i · yk ) .

k=1

That yields a very simple method for calculation of hm . For f from Section 9 this method gives h3 ≈ 24.8345, h4 ≈ 24.818. The second method. For m 6 n the values hm can be calculated another way. We are using here that f (t) = f0 + f1 t + . . . + fd td is a polynomial. The first method is technically easier then the second one. However, the second method doesn’t assume that f is a monotone decreasing function on [−1, −t0 ], and it can be applied to functions without monotonicity assumption. Let us consider Hm (y) as the symmetric polynomial Fm (t1 , . . . , tm ) in the variables ti = y·yi : Fm (t1 , . . . , tm ) = f (1)+f (−t1 )+. . .+f (−tm ). Denote by sk = sk (t1 , . . . , tm ) the power sum tk1 + . . . + tkm . Then Fm (t1 , . . . , tm ) = Ψm (s1 , . . . , sd ) = f (1) + mf0 − f1 s1 + . . . + (−1)d fd sd . From the fact that ∆m is a spherical regular simplex follows s2 = σ(s1 ) :=

z s2 + 1 − z. (m − 1)z + 1 1

(7.1)

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Any symmetric polynomial in m variables can be expressed as a polynomial of s1 , . . . , sm . Therefore, in the case k > m the power sum sk is Rk (s1 , . . . , sm ). Combining this with (7.1), we get Ψm (s1 , σ(s1 ), s3 , . . . , sd ) = Φm (s1 , s3 , . . . , sm ). Therefore, we have hm = max Φm (s1 , s3 , . . . , sm ),

(s1 , s3 , . . . , sm ) ∈ Dm ⊂ Rm−1 ,

where Dm is the domain in Rm−1 defined by the constraints ti > t0 and (7.1). Let us show now how to determine Dm for m > 2. The equation (7.1) defines the ellipsoid E : s2 = σ(s1 ) in space {t1 , . . . , tm }. Then s1 = t1 + . . . + tm attains its maximum on E atT the point with t1 = t2 = . . . = tm , and s1 achieves its minimum on E {ti > t0 } at the point with t2 = . . . = tm = t0 . From this follows w1 6 s1 6 w2 , where p (p − t20 ) (p − z 2 ) + z t0 1 + (m − 2) z w1 = + (m − 1) t0 , p = , p m−1 p w2 = m (m − 1) z + m. The equation s1 = ω gives the hyperplane, and the equation s2 = σ(ω) gives the (m − 1)-sphere in space: {(t1 , . . . , tm )}. Denote by S(ω) the (m−2)-sphere that is the intersection ofT these hyperplane and sphere. Let lk (ω) be the minimum of sk on S(ω) {ti > t0 }, and vk (ω) is its maximum. Now we have hm = max max . . . max Φm (s1 , s3 , . . . , sm ), where s1

s3

w1 6 s1 6 w2 ,

sm

lk (s1 ) 6 sk ≤ vk (s1 ), k = 3, . . . , m.

For the polynomial f from Section 9 (and Section 2) we can give more details about calculations of hm for m = 3, 4. Let us consider the case m = 3 with d = 9. In this case Fω (s3 ) = Φ3 (ω, s3 ) is a polynomial of degree 3 in the variable s3 . Lemma 4. Let f be a 9th degree polynomial f (t) = f0 + f1 t + . . . + f9 t9 such that f9 > 0, f6 = f8 = 0, and f7 > −15f9 /7. If Fω0 (s) ≤ 0 at s = l3 (ω), then the function Fω (s) achieves its maximum on the interval [l3 (ω), v3 (ω)] at s = l3 (ω).

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Proof. The expansion of s9 in terms of si1 sj2 sk3 , i + 2j + 3k = 9, is s9 =

2 3 3 7 5 1 3 s + s23 ( s31 + s2 s1 ) + s3 ( s32 − s22 s21 − s2 s41 + s61 ) + R(s1 , s2 ). 9 3 3 8 8 8 24

The coefficient of s23 s1 in s7 equals 7/9. Thus Fω (s) = −s3 f9 /9−s2 (f9 ω σ(ω)+2f9 ω 3 /3−7f7 ω/9)+sR1 (ω)+R0 (ω). Fω (s) is a cubic polynomial with negative coefficient of s3 . Then Fω (s) is a concave function for s > r, where r : Fω00 (r) = 0. Therefore, if r < l3 (ω), then Fω (s) is a concave function on the interval [l3 (ω), v3 (ω)]. r < l3 (ω) iff B(ω) := 3l3 (ω) + 6ω 3 + 9ω σ(ω) > −7ωf7 /f9 . This inequality holds for t0 < −z ≤ 0. Indeed, ω > w1 > 1 + 2z,

σ (ω) ≥ 1,

l3 (ω) > 0;

so then B(ω) > 15ω > −7ωf7 /f9 . Fω0 (l3 (ω))

6 0 implies that Fω (s) is a decreasing function The inequality on the interval [l3 (ω), v3 (ω)]. The polynomial f from Section 9 satisfies the assumptions in this lemma. Then Φ3 (ω, s) attains its maximum at the point s = l3 (ω), i.e. at the point with t1 = t2 > t3 , or with t1 > t2 ≥ t3 = t0 . If t1 = t2 > t3 , then p(ω) = Φ3 (ω, l3 (ω)) is a polynomial in ω. This polynomial is a decreasing function in the variable ω on the interval t3 > t0 . Therefore, p(ω) achieves its maximum on this interval at the point with t3 = t0 . The calculations show that for f from Section 9 h3 = max p(ω) ≈ 24.8345, when θ3 = θ0 , θ1 = θ2 ≈ 30.0715◦ . Corollary 3. Let f be the polynomial from Section 9, then h3 ≈ 24.8345. Consider the function Fω (s3 , s4 ) = Φ4 (ω, s3 , s4 ) on S(ω). Let qi ∈ S(ω) and q1 : t1 = t2 > t3 = t4 , q2 : t1 = t2 = t3 > t4 , and q3 : t1 > t2 = t3 = t4 . P i Lemma 5. Let f be a 9th degree polynomial f (t) = fi t . If f9 > 0 and f6 = f8 = 0, then the function Fω (s3 , s4 ) achieves its maximum on S(ω) with ω > 1 at one of the points (s3 (qi ), s4 (qi )), i = 1, 2, 3.

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Proof. The expansion of s9 in terms of si1 sj2 sk3 sl4 is s9 =

1 1 3 3 9 2 s s1 + s33 − s23 s31 + s4 s3 s1 + s4 s2 s31 − 16 4 9 3 4 8

3 1 − s3 s22 s21 − s3 s61 + R(s1 , s2 ). 8 24 The coefficient of s23 s1 in s7 equals 0. We have f6 = f8 = 0, then Fω (s3 , s4 ) = −f9 s9 + . . . = −f9 (s33 /9 − s23 ω 3 /3) + . . . Therefore, F33 =

∂ 2 Fω (s3 , s4 ) 2 2f9 3 2 (ω − s3 ). = −f9 ( s3 − ω 3 ) = 2 ∂ s3 3 3 3

If Fω (s3 , s4 ) has its maximum on S(ω) at the point x, and x is not a critical point of s3 on S(ω), then F33 ≤ 0. From other side, for all ti ∈ [0, 1] and s1 = ω > 1 we have s3 6 ω < ω 3 , so then F33 > 0. The function s3 on S(ω) (up to permutation of labels) has critical points at qi , i = 1, 2, 3. Corollary 4. Let f be the polynomial from Section 9, then h4 ≈ 24.818. Proof. By direct calculations it can be shown that Fω (s3 (q1 ), s4 (q1 )) > Fω (s3 (qi ), s4 (qi )) for i = 2, 3. Then Lemma 5 implies h4 = max p(ω), where p(ω) = Fω (s3 (q1 ), s4 (q1 )) = Φ4 (ω, s3 (q1 ), s4 (q1 )). The polynomial p(ω) attains its maximum h4 ≈ 24.818 at the point with θ1 = θ2 ≈ 30.2310◦ , θ3 = θ4 ≈ 51.6765◦ .

8

On calculations of h5 in four dimensions

Let us consider the case n = 4, m = 5. For simplicity here we consider only the case z = 1/2. Then δ = 60◦ and θ0 = arccos t0 < 60◦ . Denote by Γ5 the graph of the edges of ∆5 with length 60◦ , where Q5 is an optimal set. The degree of any vertex of Γ5 is not less than 3 (see Section 5). This implies that at least one vertex of Γ5 has degree 4. Indeed, if all vertices of Γ5 are of degree 3, then the sum of the degrees equals 15, i.e. is not an even number. There exists only one type of Γ5 with these conditions (Fig. 6).

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y5

y1 t P D PPP PPty3  D   D 
y  4 
 tp D   ppp D
pp D  
pp p 
α pppDp   t pDp
X XXX XXX pDppp
XD
pt y2 Fig. 6

For fixed dist(y2 , y4 ) = α, Q5 is uniquely defined up to isometry. Therefore, we have the 1-parametric family ∆5 (α) on S3 . If dist(y3 , y5 ) = β, then 2 cos α cos β + cos α + cos β = 0 (8.1) The equation (8.1) defines the function β = λ(α). Then α = λ(β), λ(90◦ ) = 90◦ . For all i we have dist(yi , e0 ) 6 θ0 , then dist(yi , yj ) 6 dist(yi , e0 ) + dist(yj , e0 ) 6 2θ0 . Suppose α 6 β, then (8.1) and the inequality β 6 2θ0 yield α0 6 α 6 90◦ 6 β 6 2θ0 ,

α0 := max{60◦ , λ(2θ0 )}.

Let H5 (y, α) = f (1) + f (−y · y1 (α)) + . . . + f (−y · y5 (α)). Then h5 = max{H5 (y, α)}, y ∈ S3 , y ·yk (α) > t0 , 1 6 k 6 5, α0 6 α 6 90◦ y,α

(8.2) We have four-dimensional optimization problem (8.2). Our first approach for this problem was to apply numerical methods [28]. For the polynomial f from Section 9 this optimization problem was solved numerically by using the Nelder-Mead simplex method: H5 (y, α) achieves

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its maximum h5 ≈ 24.6856 at α = 60◦ and y with θ1 ≈ 42.1569◦ , θ2 = θ4 ≈ 32.3025◦ , θ3 = θ5 = θ0 . (The similar approach for the case n = 4, m = 6 gives the 3-parametric family ∆6 (α, β, γ), and for f from Section 9: h6 ≈ 22.5205.) Note that (8.2) is a nonconvex constrained optimization problem. In this case, the Nelder-Mead simplex method and other local improvements methods cannot guarantee finding a global optimum. It’s possible (using estimations of derivatives) to organize computational process in such way that it gives a global optimum. However, such kind solutions are very hard to verify and some mathematicians do not accept such kind proofs. Fortunately, an estimation of h5 can be reduced to discrete optimization problems. Let dist(y1 , y) = ψ, and Φi,j (y, ψ) = f (−y · yi ) + f (−y · yj ). It’s clear that for fixed ψ, Φi,j (y, ψ) attains its maximum at some point that lies in the great 2-sphere that contains y1 , yi , yj . Now we introduce the function F (ψ, γ).6 Suppose y1 yi yj is a spherical triangle in S2 with dist(y1 , yi ) = dist(y1 , yj ) = 60◦ , dist(yi , yj ) = γ, denote by F (ψ, γ) the maximum of Φi,j (y, ψ) on S2 subject to the constraints y · yk > t0 , k = i, j. Then Φi,j (y, ψ) 6 F (ψ, γ), so then Φ2,4 (y, ψ) 6 F (ψ, α), Φ3,5 (y, ψ) 6 F (ψ, β). Thus h5 6 f (1) + f (− cos ψ) + F (ψ, α) + F (ψ, λ(α)).

(8.3)

Let α0 < α1 < . . . < αk < αk+1 = 90◦ . It’s easy to see that F (ψ, γ) is a monotone decreasing function in γ. That implies for α ∈ [αi , αi+1 ] : F (ψ, α) 6 F (ψ, αi ), F (ψ, λ(α)) 6 F (ψ, λ(αi+1 )). Therefore, from (8.3) follows h5 6 f (1) + f (− cos ψ) + max {F (ψ, αi ) + F (ψ, λ(αi+1 ))}. 06i6k

(8.4)

Note that (8.4) to reduce the dimension of the optimization problem (8.1) from 4 to 2. It is not too hard to solve this problem in general case. However, the polynomial f from Section 9 satisfies an additional assumptions that allowed to find a weak bound on h5 even more easier. Let us briefly explain how to check the following assumptions for f : 1) Φi,j (y, ψ) achieves its maximum at one of the ends of the arc ω(ψ, γ), where ω(ψ, γ) := {y : y ∈ S2 , dist(y1 , y) = ψ, y · y` > t0 , ` = i, j}; 6 F (ψ, 60◦ )

= F2 (ψ) − f (1) (see Section 2, 8, Fig. 5).

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2) F (ψ, γ) is a monotone increasing function in ψ. For given γ (γ = dist(yi , yj )) and ψ the function Φi,j (y, ψ) becomes a polynomial p(s) of degree d on [s0 , 1], where s = cos u, u = ∠yi y1 yc , and yc is the center of y1 yi yj (see Section 2, 8). Then 1) holds iff p0 (s) has no roots on (s0 , 1), either if s : p0 (s) = 0, then p00 (s) > 0. Using 1) it’s easy to check 2). For the polynomial f (t) from Section 9 if γ > 62.41◦ , then p(s) achieves its maximum at s = s0 (i.e. dist(yj , y) = θ0 ), so it’s clear that 2) holds. From other side if γ < 69.34◦ , then the arc ω(ψ, γ) lies inside the triangle y1 yi yj , therefore F (ψ, γ) increases whenever ψ increases. Note that 1) gives us the explicit expression for F (ψ, γ) = max(p(s0 ), p(1)). For fixed γ and ψ 6 ψ` from 2) follows F (ψ, γ) 6 F (ψ` , γ). Denote by ψL(i) , ψU (i) the lower and upper bounds on ψ that defined by the constraints α ∈ [αi , αi+1 ], y · yq > t0 , q = 1, . . . , 5. Let ψL(i) = ψi,0 < ψi,1 < . . . < ψi,` < ψi,`+1 = ψU (i) . Recall that f (− cos ψ) is a monotone decreasing function in ψ. Then 2) and (8.4) yield h5 6 f (1) + max max {Ri,j }, (8.5) 06i6k 06j6`

where Ri,j = f (− cos ψi,j ) + F (ψi,j+1 , αi )} + F (ψi,j+1 , λ(αi+1 )). It’s very easy to apply this method. Here we need just to calculate the matrix (Ri,j ) and the maximal value of its entries gives the bound on h5 . For f from Section 9 and t0 ≈ 0.60794, θ0 = arccos t0 ≈ 52.5588◦ , f (−t0 ) = 0, this method gives the bound h5 < 24.8434.7 Now we show how to find an upper bound on h6 . Let {e0 , y1 , . . . , y6 } ∈ S3 , H(y1 , . . . , y6 ) = f (1) + f (− cos θ1 ) + . . . + f (− cos θ6 ), where θi = dist(e0 , yi ). Suppose θ1 6 θ2 6 . . . 6 θ6 . Now we prove that θ6 > 45◦ . That can be proven as Corollary all θi < 45◦ . √ 2 (Section 4). Conversely, 2 In this case t0∗ = cos θ6 > 1/ 2, and ω = arccos [(1/2 − t0∗ )/(1 − t20∗ )] > 90◦ . But if u > 90◦ , then A(3, ω) 6 4 (see [32, 18]) - a contradiction. (In fact we proved that θ5 > 45◦ also.) 7 R achieves its maximum at α = 60◦ , ψ ≈ 30.9344◦ . Note that this bound exceeds the tight bound on h5 ≈ 24.6856 given by numerical methods.

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Let us consider two cases: (i) θ0 > θ6 > 50◦ (ii) 50◦ > θ6 > 45◦ . (i) H(y1 , . . . , y6 ) = H(y1 , . . . , y5 ) + f (− cos θ6 ). We have H(y1 , . . . , y5 ) 6 h5 < 24.8434,

f (− cos θ6 ) < f (− cos 50◦ ) ≈ 0.0906,

then H(y1 , . . . , y6 ) < 24.934. (ii) In this case all θi 6 50◦ . Therefore, we can apply (8.5) for θ0 = 50◦ . This method gives h5 (50◦ ) < 23.9181, then H(y1 , . . . , y5 ) 6 h5 (50◦ ) < 23.9181, so then H(y1 , . . . , y6 ) < 23.9181 + f (− cos 45◦ ) ≈ 23.9181 + 0.4533 = 24.3714 Thus h6 < max{24.934, 24.3714} = 24.934

9

k(4) = 24

For n = 4, z = cos 60◦ = 1/2 we apply this extension of Delsarte’s method with f (t) = 53.76t9 − 107.52t7 + 70.56t5 + 16.384t4 − −9.832t3 − 4.128t2 − 0.434t − 0.016 (4)

The expansion of f in terms of Uk = Gk is f = U0 + 2U1 + 6.12U2 + 3.484U3 + 5.12U4 + 1.05U9 The polynomial f has two roots on [−1, 1]: t1 = −t0 , t0 ≈ 0.60794, t2 = 1/2, f (t) 6 0 for t ∈ [−t0 , 1/2], and f is a monotone decreasing function on the interval [−1, −t0 ]. The last property holds because there are no zeros of the derivative f 0 (t) on [−1, −t0 ]. Therefore, f satisfies (∗) for z = 1/2. Remark. The polynomial f was found by using the algorithm in Section 6. This algorithm for n = 4, z = 1/2, d = 9, N = 2000, t0 = 0.6058 gives E ≈ 24.7895. For the polynomial f the coefficients ck were changed to “better looking” ones with E ≈ 24.8644.

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6 5 4 3 2 1 0 −1 −1

−0.8 −0.6 −0.4 −0.2

0

0.2

0.4

0.6

0.8

Fig. 7. The graph of the function f (t) We have t0 > 0.6058. Then Corollary 2 gives µ 6 6. Consider all m 6 6. h0 = f (1) = 18.774, h2 = f (1) +

h1 = f (1) + f (−1) = 24.48.

max {(f (− cos θ) + f (− cos(60◦ − θ))} ≈ 24.8644,

30◦ 6θ6θ0

where θ0 = arccos t0 ≈ 52.5588◦ . Note that h2 can be calculated by the same method as in Section 2. Here h2 = f (1) + 2f (− cos 30◦ ) also. In Sections 7, 8 have been shown that h3 ≈ 24.8345, Theorem 4.

h4 ≈ 24.818,

h5 < 24.8434,

h6 < 24.934.

k(4) = 24

Proof. Let X be a spherical 1/2-code in S3 with M = k(4) points. The polynomial f is such that hmax < 25, then combining this and Theorem 2, we get k(4) 6 hmax < 25. Recall that k(4) > 24. Consequently, k(4) = 24.

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193

Concluding remarks

The algorithm in Section 6 can be applied to other dimensions and spherical z-codes. If t0 = 1, then the algorithm gives the Delsarte method. E is an estimation of hmax in this algorithm. Direct application of the method developed in this paper, presumably could lead to some improvements in the upper bounds on kissing numbers in dimensions 9, 10, 16, 17, 18 given in [11, Table 1.5]. (“Presumably” because the equality hmax = E is not proven yet.) In 9 and 10 dimensions Table 1.5 gives: 306 6 k(9) 6 380, 500 6 k(10) 6 595. The algorithm gives: n = 9 : deg f = 11, E = h1 = 366.7822, t0 = 0.54; n = 10 : deg f = 11, E = h1 = 570.5240, t0 = 0.586. For these dimensions there is a good chance to prove that k(9) 6 366, k(10) 6 570. From the equality k(3) = 12 follows ϕ3 (13) < 60◦ . The method gives ϕ3 (13) < 59.4◦ (deg f = 11). The lower bound on ϕ3 (13) is 57.1367◦ [18]. Therefore, we have 57.1367◦ 6 ϕ3 (13) < 59.4◦ . The method gives ϕ4 (25) < 59.81◦ , ϕ4 (24) < 60.5◦ . (This is theorem that can be proven by the same method as Theorem 4.) That improve the bounds: ϕ4 (25) < 60.79◦ , ϕ4 (24) < 61.65◦ [25] (cf. [6]); ϕ4 (24) < 61.47◦ [6]; ϕ4 (25) < 60.5◦ ,

ϕ4 (24) < 61.41◦ [4].

Now in these cases we have 57.4988◦ < ϕ4 (25) < 59.81◦ ,

60◦ 6 ϕ4 (24) < 60.5◦ .

For all cases that were considered (z 6 0.6) this method gives better bounds than Fejes T´ oth’s bounds for ϕ3 (M ) [18] and Coxeter’s bounds for all ϕn (M ) [12]. However, for n = 5, 6, 7 direct use of this generalization of the Delsarte method does not give better upper bounds on k(n) than the Delsarte method. It is an interesting problem to find better methods. Acknowledgment. I wish to thank Eiichi Bannai, Ivan Dynnikov, Dmitry Leshchiner, Sergei Ovchinnikov, Makoto Tagami and G¨ unter Ziegler for helpful discussions and useful comments on this paper.

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References [1] M. Aigner and G.M. Ziegler, Proofs from THE BOOK, Springer, 1998 (first ed.) and 2002 (second ed.) [2] K. Anstreicher, The thirteen spheres: A new proof, Discrete and Computational Geometry, 31(2004), 613-625. [3] V.V. Arestov and A.G. Babenko, On Delsarte scheme of estimating the contact numbers, Trudy Mat. Inst. im. V.A.Steklova 219, 1997, 44-73; English translation, Proc. of the Steklov Inst. of Math. 219, 1997, 36-65. [4] V.V. Arestov and A.G. Babenko, On kissing number in four dimensions, in Proc. Conf. memory of Paul Erd¨os, Budapest, Hingary, July 4-11, 1999, A.Sali, M.Simonovits and V.T.S´os (eds), J. Bolyai Math. Soc., Budapest, 1999, 10-14. [5] P. Boyvalenkov, S. Dodunekov and O. R. Musin, A survey on the kissing numbers, Serdica Mathematical Journal, 38:4 (2012), 507522. [6] P.G. Boyvalenkov, D.P. Danev and S.P. Bumova, Upper bounds on the minimum distance of spherical codes, IEEE Trans. Inform. Theory, 42(5), 1996, 1576-1581. [7] K. B¨ or¨ oczky, Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hung. 32 (1978), 243-261. [8] K. B¨ or¨ oczky, The Newton-Gregory problem revisited, Proc. Discrete Geometry, Marcel Dekker, 2003, 103-110. [9] B.C. Carlson, Special functions of applied mathematics, Academic Press, 1977. [10] B. Casselman, The difficulties of kissing in three dimensions, Notices Amer. Math. Soc., 51(2004), 884-885. [11] J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices, and Groups, New York, Springer-Verlag, 1999 (Third Edition).

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[12] H.S.M. Coxeter, An upper bound for the number of equal nonoverlapping spheres that can touch another of the same size, Proc. of Symp. in Pure Math. AMS, 7 (1963), 53-71 = Chap. 9 of H.S.M. Coxeter, Twelve Geometric Essays, Southern Illinois Press, Carbondale Il, 1968. [13] L. Danzer, Finite point-sets on S2 with minimum distance as large as possible, Discr. Math., 60 (1986), 3-66. [14] L. Danzer, B. Gr¨ unbaum, and V. Klee. Helly’s theorem and its relatives. Proc. Sympos. Pure Math., vol. 7, AMS, Providence, RI, 1963, pp. 101-180. [15] Ph. Delsarte, Bounds for unrestricted codes by linear programming, Philips Res. Rep., 27, 1972, 272-289. [16] Ph. Delsarte, J.M. Goethals and J.J. Seidel, Spherical codes and designs, Geom. Dedic., 6, 1977, 363-388. [17] A. Erd´elyi, editor, Higher Transcendental Function, McGraw-Hill, NY, 3 vols, 1953, Vol. II, Chap. XI. [18] L. Fejes T´ oth, Lagerungen in der Ebene, auf der Kugel und in Raum, Springer-Verlag, 1953; Russian translation, Moscow, 1958. [19] T. Hales, The status of the Kepler conjecture, Mathematical Intelligencer 16(1994), 47-58. [20] R. Hoppe, Bemerkung der Redaction, Archiv Math. Physik (Grunet) 56 (1874), 307-312. [21] W.-Y. Hsiang, The geometry of spheres, in Differential Geometry (Shanghai,1991), Word Scientific, River Edge, NJ, 1993, pp. 92-107. [22] W.-Y. Hsiang, Least Action Principle of Crystal Formation of Dense Packing Type and Kepler’s Conjecture, World Scientific, 2001. [23] G.A. Kabatiansky and V.I. Levenshtein, Bounds for packings on a sphere and in space, Problemy Peredachi informacii 14(1), 1978, 3-25; English translation, Problems of Information Transmission, 14(1), 1978, 1-17.

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[24] J. Leech, The problem of the thirteen spheres, Math. Gazette 41 (1956), 22-23. [25] V.I. Levenshtein, On bounds for packing in n-dimensional Euclidean space, Sov. Math. Dokl. 20(2), 1979, 417-421. [26] O.R. Musin, The kissing number in four dimensions, Annals of Mathematics, 168 (2008), No. 1, 1-32 [27] Oleg R. Musin, The kissing problem in three dimensions, Discrete & Computational Geometry 35 (2006), no. 3, 375-384 [28] O.R. Musin, The problem of the twenty-five spheres, Russian Math. Surveys, 58(2003), 794-795 [29] A.M. Odlyzko and N.J.A. Sloane, New bounds on the number of unit spheres that can touch a unit sphere in n dimensions, J. of Combinatorial Theory A26(1979), 210-214. [30] F. Pfender and G.M. Ziegler, Kissing numbers, sphere packings, and some unexpected proofs, Notices Amer. Math. Soc., 51(2004), 873-883. [31] I.J. Schoenberg, Positive definite functions on spheres, Duke Math. J., 9 (1942), 96-107. [32] K. Sch¨ utte and B.L. van der Waerden, Auf welcher Kugel haben 5,6,7,8 oder 9 Punkte mit Mindestabstand 1 Platz? Math. Ann. 123 (1951), 96-124. [33] K. Sch¨ utte and B.L. van der Waerden, Das Problem der dreizehn Kugeln, Math. Ann. 125 (1953), 325-334. [34] G.G. Szpiro, Kepler’s conjecture, Wiley, 2002. [35] G.G. Szpiro, Newton and the kissing problem, http://plus.maths.org/issue23/features/kissing/ [36] A.D. Wyner, Capabilities of bounded discrepancy decoding, Bell Sys. Tech. J. 44 (1965), 1061-1122.

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First Yaroslavl Summer School on Discrete and Computational Geometry July – August, 2012 Lecture Notes

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