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15 BASIC PROPERTIES of CONVEX POLYTOPES Martin Henk, J¨Urgenrichter-Gebert, and G¨Unterm
15 BASIC PROPERTIES OF CONVEX POLYTOPES Martin Henk, J¨urgenRichter-Gebert, and G¨unterM. Ziegler INTRODUCTION Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their im- portance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry to linear and combinatorial optimiza- tion. In this chapter we try to give a short introduction, provide a sketch of \what polytopes look like" and \how they behave," with many explicit examples, and briefly state some main results (where further details are given in subsequent chap- ters of this Handbook). We concentrate on two main topics: • Combinatorial properties: faces (vertices, edges, . , facets) of polytopes and their relations, with special treatments of the classes of low-dimensional poly- topes and of polytopes \with few vertices;" • Geometric properties: volume and surface area, mixed volumes, and quer- massintegrals, including explicit formulas for the cases of the regular simplices, cubes, and cross-polytopes. We refer to Gr¨unbaum [Gr¨u67]for a comprehensive view of polytope theory, and to Ziegler [Zie95] respectively to Gruber [Gru07] and Schneider [Sch14] for detailed treatments of the combinatorial and of the convex geometric aspects of polytope theory. 15.1 COMBINATORIAL STRUCTURE GLOSSARY d V-polytope: The convex hull of a finite set X = fx1; : : : ; xng of points in R , n n X i X P = conv(X) := λix λ1; : : : ; λn ≥ 0; λi = 1 : i=1 i=1 H-polytope: The solution set of a finite system of linear inequalities, d T P = P (A; b) := x 2 R j ai x ≤ bi for 1 ≤ i ≤ m ; with the extra condition that the set of solutions is bounded, that is, such that m×d there is a constant N such that jjxjj ≤ N holds for all x 2 P . -
Report No. 17/2005
Mathematisches Forschungsinstitut Oberwolfach Report No. 17/2005 Discrete Geometry Organised by Martin Henk (Magdeburg) Jiˇr´ıMatouˇsek (Prague) Emo Welzl (Z¨urich) April 10th – April 16th, 2005 Abstract. The workshop on Discrete Geometry was attended by 53 par- ticipants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Introduction by the Organisers The Discrete Geometry workshop was attended by 53 participants from a wide range of geographic regions, many of them young researchers (some supported by a grant from the European Union). The morning sessions consisted of survey talks providing an overview of recent developments in Discrete Geometry: Extremal problems concerning convex lattice polygons. (Imre B´ar´any) • Universally optimal configurations of points on spheres. (Henry Cohn) • Polytopes, Lie algebras, computing. (Jes´us A. De Loera) • On incidences in Euclidean spaces. (Gy¨orgy Elekes) • Few-distance sets in d-dimensional normed spaces. (Zolt´an F¨uredi) • On norm maximization in geometric clustering. (Peter Gritzmann) • Abstract regular polytopes: recent developments. (Peter McMullen) • Counting crossing-free configurations in the plane. (Micha Sharir) • Geometry in additive combinatorics. (J´ozsef Solymosi) • Rigid components: geometric problems, combinatorial solutions. (Ileana • Streinu) Forbidden patterns. (J´anos Pach) • 926 Oberwolfach Report 17/2005 Projected polytopes, Gale diagrams, and polyhedral surfaces. (G¨unter M. • Ziegler) What is known about unit cubes? (Chuanming Zong) • There were 16 shorter talks in the afternoon, an open problem session chaired by Jes´us De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cin- derella (J¨urgen Richter-Gebert). -
Arxiv:Math/0311272V2 [Math.MG] 11 Jun 2004
Hyperbolic Coxeter n-polytopes with n +3 facets P. Tumarkin Abstract. A polytope is called a Coxeter polytope if its dihedral angles are integer parts of π. In this paper we prove that if a non- compact Coxeter polytope of finite volume in IHn has exactly n+3 facets then n ≤ 16. We also find an example in IH16 and show that it is unique. 1. Consider a convex polytope P in n-dimensional hyperbolic space IHn. A polytope is called a Coxeter polytope if its dihedral angles are integer parts of π. Any Coxeter polytope P is a fundamental domain of the discrete group generated by the reflections with respect to facets of P . Of special interest are hyperbolic Coxeter polytopes of finite volume. In contrast to spherical and Euclidean cases there is no complete classification of such polytopes. It is known that dimension of compact Coxeter polytope does not exceed 29 (see [12]), and dimension of non-compact polytope of finite volume does not exceed 995 (see [9]). Coxeter polytopes in IH 3 are completely characterized by Andreev [1], [2]. There exists a complete classification of hyperbolic simplices [8], [11] and hyperbolic n-polytopes with n + 2 facets [7] (see also [13]), [5], [10]). In [5] Esselmann proved that n-polytopes with n + 3 facets do not exist in IHn, n> 8, and the example found by Bugaenko [3] in IH8 is unique. There is an example of finite volume non-compact polytope in IH15 with 18 facets (see [13]). The main result of this note is the following theorem: Theorem 1. -
Combinatorial Aspects of Convex Polytopes Margaret M
Combinatorial Aspects of Convex Polytopes Margaret M. Bayer1 Department of Mathematics University of Kansas Carl W. Lee2 Department of Mathematics University of Kentucky August 1, 1991 Chapter for Handbook on Convex Geometry P. Gruber and J. Wills, Editors 1Supported in part by NSF grant DMS-8801078. 2Supported in part by NSF grant DMS-8802933, by NSA grant MDA904-89-H-2038, and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center, NSF-STC88-09648. 1 Definitions and Fundamental Results 3 1.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.2 Faces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.3 Polarity and Duality : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.4 Overview : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2 Shellings 4 2.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 Euler's Relation : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Line Shellings : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.4 Shellable Simplicial Complexes : : : : : : : : : : : : : : : : : : : 5 2.5 The Dehn-Sommerville Equations : : : : : : : : : : : : : : : : : : 6 2.6 Completely Unimodal Numberings and Orientations : : : : : : : 7 2.7 The Upper Bound Theorem : : : : : : : : : : : : : : : : : : : : : 8 2.8 The Lower Bound Theorem : : : : : : : : : : : : : : : : : : : : : 9 2.9 Constructions Using Shellings : : : : : : : : : : : : : -
10 GEOMETRIC GRAPH THEORY J´Anos Pach
10 GEOMETRIC GRAPH THEORY J´anos Pach INTRODUCTION In the traditional areas of graph theory (Ramsey theory, extremal graph theory, random graphs, etc.), graphs are regarded as abstract binary relations. The relevant methods are often incapable of providing satisfactory answers to questions arising in geometric applications. Geometric graph theory focuses on combinatorial and geometric properties of graphs drawn in the plane by straight-line edges (or, more generally, by edges represented by simple Jordan arcs). It is a fairly new discipline abounding in open problems, but it has already yielded some striking results that have proved instrumental in the solution of several basic problems in combinatorial and computational geometry (including the k-set problem and metric questions discussed in Sections 1.1 and 1.2, respectively, of this Handbook). This chapter is partitioned into extremal problems (Section 10.1), crossing numbers (Section 10.2), and generalizations (Section 10.3). 10.1 EXTREMAL PROBLEMS Tur´an’s classical theorem [Tur54] determines the maximum number of edges that an abstract graph with n vertices can have without containing, as a subgraph, a complete graph with k vertices. In the spirit of this result, one can raise the follow- ing general question. Given a class of so-called forbidden geometric subgraphs, what is the maximum number of edgesH that a geometric graph of n vertices can have without containing a geometric subgraph belonging to ? Similarly, Ramsey’s the- orem [Ram30] for abstract graphs has some natural analoguesH for geometric graphs. In this section we will be concerned mainly with problems of these two types. -
GEOCHEMICAL PHASE DIAGRAMS and GALE DIAGRAMS Contents 1
GEOCHEMICAL PHASE DIAGRAMS AND GALE DIAGRAMS P.H. EDELMAN, S.W. PETERSON, V. REINER, AND J.H. STOUT Abstract. The problem of predicting the possible topologies of a geochemical phase diagram, based on the chemical formula of the phases involved, is shown to be intimately connected with and aided by well-studied notions in discrete geometry: Gale diagrams, triangulations, secondary fans, and oriented matroids. Contents 1. Introduction 2 2. A geochemistry-discrete geometry dictionary 6 3. Chemical composition space 6 4. Triangulations and subdivisions 10 5. Secondary fans 13 6. Gale diagrams and duality 17 7. Geometry of the phase diagram in general 21 7.1. Bistellar operations and 1-dimensional stability fields 23 7.2. Invariant points and indifferent crossings 25 8. The case m = n + 2: phase diagram = Gale diagram 28 9. The case m = n + 3: phase diagram = affine Gale diagram 30 9.1. The spherical representation: closed nets 31 9.2. Two-dimensional affine Gale diagrams 33 10. Further implications/applications 35 10.1. Slopes around invariant points 35 10.2. Counting potential solutions 37 References 40 1991 Mathematics Subject Classification. 52B35, 52C40, 86A99. Key words and phrases. Gale diagram, Gale transform, phase diagram, trian- gulation, secondary polytope, secondary fan, geochemistry, heterogeneous equilib- rium, closed net, Euler sphere. Work of second author was carried out partly as a Masters Thesis at the Uni- versity of Minnesota Center for Industrial Mathematics, and partly supported by a University of Minnesota Grant-in-Aid of Research. Third author was partially supported by NSF grant DMS{9877047. 1 2 P.H. -
On Zonotopes
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 159, September 1971 ON ZONOTOPES BY P. McMULLEN Abstract. In this paper is described a diagram technique for zonotopes, or vector sums of line segments, which is analogous to that of Gale diagrams for general polytopes, and central diagrams for centrally symmetric polytopes. The use of these new zonal diagrams leads to relationships between zonotopes with n zones of dimen- sions d and n —d, and enables one to enumerate all the combinatorial types of d- zonotopes with ntsd+2 zones. The connexion between arrangements of hyperplanes in projective space and zonotopes leads to corresponding new results about arrangements. 1. Introduction. Several recent advances in the theory of convex polytopes have stemmed from the idea of Gale diagrams, which were first conceived by Gale [1956], and later much elaborated by Perles (see Grünbaum [1967, 5.4, 6.3 and elsewhere]). The technique is to represent a o'-polytope with n vertices by a diagram of n points in (« —d— l)-dimensional space, from which all the combinatorial properties of the original polytope can be determined. However, Gale diagrams proved ineffective in dealing with the problems of centrally symmetric polytopes. McMullen-Shephard [1968] described the appro- priate analogues of Gale diagrams for centrally symmetric polytopes ; these dia- grams are known as central diagrams or c.s. diagrams. One subclass of the centrally symmetric polytopes, which is interesting for many reasons (in particular, the connexion with arrangements of hyperplanes, for which see §7 below), comprises the zonotopes. Since a ¿-dimensional zonotope has at least 2" vertices, even central diagrams fail to give much information about zono- topes. -
Random Gale Diagrams and Neighborly Polytopes in High Dimensions
Random Gale diagrams and neighborly polytopes in high dimensions Rolf Schneider Abstract Taking up a suggestion of David Gale from 1956, we generate sets of combinatorially isomorphic polytopes by choosing their Gale diagrams at random. We find that in high dimensions, and under suitable assumptions on the growth of the involved parameters, the obtained polytopes have strong neighborliness properties, with high probability. Keywords: Gale diagram, random polytope, neighborly polytope, high dimensions 2020 Mathematics Subject Classification: Primary 52B35, Secondary 60D05 1 Introduction The purpose of the following is, roughly speaking, to introduce a new class of random poly- topes, which have strong neighborliness properties in high dimensions, with overwhelming probability. The main idea is to generate not directly the polytopes by a random procedure, but their Gale diagrams. d Let us first recall that a convex polytope P in Euclidean space R is called k-neighborly if any k or fewer vertices of P are neighbors, which means that their convex hull is a face of P . A bd=2c-neighborly polytope is called neighborly. It is known that a k-neighborly d-polytope with k > bd=2c must be a simplex, and that in dimensions d ≥ 4 there are neighborly polytopes with any number of vertices. It appears that neighborliness properties (though not under this name) of certain polytopes were first noted by Carath´eodory [2], and that the proper investigation of neighborly polytopes began with the work of Gale [7]. For more information on neighborly polytopes (for example, their important role in the Upper bound theorem), we refer to the books by Gr¨unbaum [8], McMullen and Shephard [12], Ziegler [19], Matouˇsek[10]. -
Arxiv:2009.07252V2 [Math.CO] 19 Sep 2020 Ol Oase H Oiaigqeto.W Reydsrb He E Three Describe Briefly We Defi Question
MINKOWSKI SUMMANDS OF CUBES FEDERICO CASTILLO, JOSEPH DOOLITTLE, BENNET GOECKNER, MICHAEL S. ROSS, AND LI YING Abstract. In pioneering works of Meyer and of McMullen in the early 1970s, the set of Minkowski summands of a polytope was shown to be a polyhedral cone called the type cone. Explicit computations of type cones are in general intractable. Nevertheless, we show that the type cone of the product of simplices is simplicial. This remarkably simple result derives from insights about rainbow point configurations and the work of McMullen. 1. Introduction A fundamental operation on polytopes is Minkowski addition. In this paper, we consider the reverse of this operation. Our motivating question is “Given a polytope P, what can we say about the set of its Minkowski summands?” It is convenient to modify this question and consider the set TMink(P) of weak Minkowski summands, polytopes that are summands of some positive dilate of P, up to translation equivalence. With this perspective, there are multiple equivalent definitions that provide tools to answer the motivating question. We briefly describe three existing techniques to parametrize the set TMink(P) as pointed polyhedral cone, which we refer to as the type cone of P. The starting point is a theorem of Shephard [13, Section 15] characterizing the weak Minkowski summands in terms of their support functions. In [21], Meyer used this connec- tion to give a parametrization of TMink(P) using one parameter for each facet, so we refer to this construction as the facet parametrization. In the context of algebraic geometry, the type cone is the nef cone of the toric variety associated to P, when P is a rational poly- tope. -
Arxiv:1908.05503V2 [Math.AG] 14 Sep 2020 D We Are Interested by the Existence of the Positive Solutions of (1.1) in R>0
SIGN CONDITIONS FOR THE EXISTENCE OF AT LEAST ONE POSITIVE SOLUTION OF A SPARSE POLYNOMIAL SYSTEM FRED´ ERIC´ BIHAN, ALICIA DICKENSTEIN, AND MAGAL´I GIAROLI Abstract. We give sign conditions on the support and coefficients of a sparse system of d generalized polynomials in d variables that guar- antee the existence of at least one positive real root, based on degree theory and Gale duality. In the case of integer exponents, we relate our sufficient conditions to algebraic conditions that emerged in the study of toric ideals. 1. Introduction Deciding whether a real polynomial system has a positive solution is a basic question, that is decidable via effective elimination of quantifiers [1]. There are few results on lower bounds on the number of real or positive roots of polynomial systems (see e.g. [3, 18, 19, 23]). In this paper, we consider generalized polynomial systems, that is, polynomials with real exponents, for which the positive solutions are well defined. We give sign conditions on the support and on the coefficients of a sparse system of d generalized poly- nomials in d variables that guarantee the existence of at least one positive real root, based on degree theory and Gale duality. d We fix an exponent set A = fa1; : : : ; ang ⊂ R of cardinality n and for d×n any given real matrix C = (cij) 2 R we consider the associated sparse generalized multivariate polynomial system in d variables x = (x1; : : : ; xd) with support A: n X aj (1.1) fi(x) = cijx = 0 ; i = 1; : : : ; d: j=1 arXiv:1908.05503v2 [math.AG] 14 Sep 2020 d We are interested by the existence of the positive solutions of (1.1) in R>0. -
C Copyright 2019 Amy Wiebe Realization Spaces of Polytopes and Matroids
c Copyright 2019 Amy Wiebe Realization spaces of polytopes and matroids Amy Wiebe A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2019 Reading Committee: Rekha R. Thomas, Chair Isabella Novik Jarod Alper Program Authorized to Offer Degree: Mathematics University of Washington Abstract Realization spaces of polytopes and matroids Amy Wiebe Chair of the Supervisory Committee: Dr Rekha R. Thomas Mathematics Chapter 1 describes several models for the realization space of a polytope. These models include the classical model, a model representing realizations in the Grassmannian, a new model which represents realizations by slack matrices, and a model which represents poly- topes by their Gale transforms. We explore the connections between these models, and show how they can be exploited to obtain useful parametrizations of the slack realization space. Chapter 2 introduces a natural model for the realization space of a polytope up to projec- tive equivalence which we call the slack realization space of the polytope. The model arises from the positive part of an algebraic variety determined by the slack ideal of the polytope. This is a saturated determinantal ideal that encodes the combinatorics of the polytope. The slack ideal offers an effective computational framework for several classical questions about polytopes such as rational realizability, non-prescribability of faces, and realizability of combinatorial polytopes. Chapter 3 studies the simplest possible slack ideals, which are toric, and explores their connections to projectively unique polytopes. We prove that if a projectively unique polytope has a toric slack ideal, then it is the toric ideal of the bipartite graph of vertex-facet non- incidences of the polytope. -
Polytopes Course Notes
Polytopes Course Notes Carl W. Lee Department of Mathematics University of Kentucky Lexington, KY 40506 [email protected] Fall 2013 i Contents 1 Polytopes 1 1.1 Convex Combinations and V-Polytopes . 1 1.2 Linear Inequalities and H-Polytopes . 8 1.3 H-Polytopes are V-Polytopes . 10 1.4 V-Polytopes are H-Polytopes . 13 2 Theorems of the Alternatives 14 2.1 Systems of Equations . 14 2.2 Fourier-Motzkin Elimination | A Starting Example . 16 2.3 Showing our Example is Feasible . 19 2.4 An Example of an Infeasible System . 20 2.5 Fourier-Motzkin Elimination in General . 22 2.6 More Alternatives . 24 2.7 Exercises: Systems of Linear Inequalities . 25 3 Faces of Polytopes 28 3.1 More on Vertices . 28 3.2 Faces . 30 3.3 Polarity and Duality . 32 4 Some Polytopes 36 4.1 Regular and Semiregular Polytopes . 36 4.2 Polytopes in Combinatorial Optimization . 38 5 Representing Polytopes 42 5.1 Schlegel Diagrams . 42 5.2 Gale Transforms and Diagrams . 42 6 Euler's Relation 44 6.1 Euler's Relation for 3-Polytopes . 44 6.2 Some Consequences of Euler's Relation for 3-Polytopes . 46 6.3 Euler's Relation in Higher Dimensions . 48 6.4 Gram's Theorem . 51 7 The Dehn-Sommerville Equations 52 7.1 3-Polytopes . 52 7.2 Simple Polytopes . 52 ii 7.3 Simplicial Polytopes . 54 d 7.4 The Affine Span of f(Ps ) ............................ 57 7.5 Vertex Figures . 58 7.6 Notes . 60 8 Shellability 61 9 The Upper Bound Theorem 66 10 The Lower Bound Theorem 71 10.1 Stacked Polytopes .