Volumes and Integrals Over Polytopes
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
A General Geometric Construction of Coordinates in a Convex Simplicial Polytope
A general geometric construction of coordinates in a convex simplicial polytope ∗ Tao Ju a, Peter Liepa b Joe Warren c aWashington University, St. Louis, USA bAutodesk, Toronto, Canada cRice University, Houston, USA Abstract Barycentric coordinates are a fundamental concept in computer graphics and ge- ometric modeling. We extend the geometric construction of Floater’s mean value coordinates [8,11] to a general form that is capable of constructing a family of coor- dinates in a convex 2D polygon, 3D triangular polyhedron, or a higher-dimensional simplicial polytope. This family unifies previously known coordinates, including Wachspress coordinates, mean value coordinates and discrete harmonic coordinates, in a simple geometric framework. Using the construction, we are able to create a new set of coordinates in 3D and higher dimensions and study its relation with known coordinates. We show that our general construction is complete, that is, the resulting family includes all possible coordinates in any convex simplicial polytope. Key words: Barycentric coordinates, convex simplicial polytopes 1 Introduction In computer graphics and geometric modelling, we often wish to express a point x as an affine combination of a given point set vΣ = {v1,...,vi,...}, x = bivi, where bi =1. (1) i∈Σ i∈Σ Here bΣ = {b1,...,bi,...} are called the coordinates of x with respect to vΣ (we shall use subscript Σ hereafter to denote a set). In particular, bΣ are called barycentric coordinates if they are non-negative. ∗ [email protected] Preprint submitted to Elsevier Science 3 December 2006 v1 v1 x x v4 v2 v2 v3 v3 (a) (b) Fig. -
7 LATTICE POINTS and LATTICE POLYTOPES Alexander Barvinok
7 LATTICE POINTS AND LATTICE POLYTOPES Alexander Barvinok INTRODUCTION Lattice polytopes arise naturally in algebraic geometry, analysis, combinatorics, computer science, number theory, optimization, probability and representation the- ory. They possess a rich structure arising from the interaction of algebraic, convex, analytic, and combinatorial properties. In this chapter, we concentrate on the the- ory of lattice polytopes and only sketch their numerous applications. We briefly discuss their role in optimization and polyhedral combinatorics (Section 7.1). In Section 7.2 we discuss the decision problem, the problem of finding whether a given polytope contains a lattice point. In Section 7.3 we address the counting problem, the problem of counting all lattice points in a given polytope. The asymptotic problem (Section 7.4) explores the behavior of the number of lattice points in a varying polytope (for example, if a dilation is applied to the polytope). Finally, in Section 7.5 we discuss problems with quantifiers. These problems are natural generalizations of the decision and counting problems. Whenever appropriate we address algorithmic issues. For general references in the area of computational complexity/algorithms see [AB09]. We summarize the computational complexity status of our problems in Table 7.0.1. TABLE 7.0.1 Computational complexity of basic problems. PROBLEM NAME BOUNDED DIMENSION UNBOUNDED DIMENSION Decision problem polynomial NP-hard Counting problem polynomial #P-hard Asymptotic problem polynomial #P-hard∗ Problems with quantifiers unknown; polynomial for ∀∃ ∗∗ NP-hard ∗ in bounded codimension, reduces polynomially to volume computation ∗∗ with no quantifier alternation, polynomial time 7.1 INTEGRAL POLYTOPES IN POLYHEDRAL COMBINATORICS We describe some combinatorial and computational properties of integral polytopes. -
1 Lifts of Polytopes
Lecture 5: Lifts of polytopes and non-negative rank CSE 599S: Entropy optimality, Winter 2016 Instructor: James R. Lee Last updated: January 24, 2016 1 Lifts of polytopes 1.1 Polytopes and inequalities Recall that the convex hull of a subset X n is defined by ⊆ conv X λx + 1 λ x0 : x; x0 X; λ 0; 1 : ( ) f ( − ) 2 2 [ ]g A d-dimensional convex polytope P d is the convex hull of a finite set of points in d: ⊆ P conv x1;:::; xk (f g) d for some x1;:::; xk . 2 Every polytope has a dual representation: It is a closed and bounded set defined by a family of linear inequalities P x d : Ax 6 b f 2 g for some matrix A m d. 2 × Let us define a measure of complexity for P: Define γ P to be the smallest number m such that for some C s d ; y s ; A m d ; b m, we have ( ) 2 × 2 2 × 2 P x d : Cx y and Ax 6 b : f 2 g In other words, this is the minimum number of inequalities needed to describe P. If P is full- dimensional, then this is precisely the number of facets of P (a facet is a maximal proper face of P). Thinking of γ P as a measure of complexity makes sense from the point of view of optimization: Interior point( methods) can efficiently optimize linear functions over P (to arbitrary accuracy) in time that is polynomial in γ P . ( ) 1.2 Lifts of polytopes Many simple polytopes require a large number of inequalities to describe. -
The Orientability of Small Covers and Coloring Simple Polytopes
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Osaka City University Repository Nakayama, H. and Nishimura, Y. Osaka J. Math. 42 (2005), 243–256 THE ORIENTABILITY OF SMALL COVERS AND COLORING SIMPLE POLYTOPES HISASHI NAKAYAMA and YASUZO NISHIMURA (Received September 1, 2003) Abstract Small Cover is an -dimensional manifold endowed with a Z2 action whose or- bit space is a simple convex polytope . It is known that a small cover over is characterized by a coloring of which satisfies a certain condition. In this paper we shall investigate the topology of small covers by the coloring theory in com- binatorics. We shall first give an orientability condition for a small cover. In case = 3, an orientable small cover corresponds to a four colored polytope. The four color theorem implies the existence of orientable small cover over every simple con- vex 3-polytope. Moreover we shall show the existence of non-orientable small cover over every simple convex 3-polytope, except the 3-simplex. 0. Introduction “Small Cover” was introduced and studied by Davis and Januszkiewicz in [5]. It is a real version of “Quasitoric manifold,” i.e., an -dimensional manifold endowed with an action of the group Z2 whose orbit space is an -dimensional simple convex poly- tope. A typical example is provided by the natural action of Z2 on the real projec- tive space R whose orbit space is an -simplex. Let be an -dimensional simple convex polytope. Here is simple if the number of codimension-one faces (which are called “facets”) meeting at each vertex is , equivalently, the dual of its boundary complex ( ) is an ( 1)-dimensional simplicial sphere. -
Combinatorial and Discrete Problems in Convex Geometry
COMBINATORIAL AND DISCRETE PROBLEMS IN CONVEX GEOMETRY A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Matthew R. Alexander September, 2017 Dissertation written by Matthew R. Alexander B.S., Youngstown State University, 2011 M.A., Kent State University, 2013 Ph.D., Kent State University, 2017 Approved by Dr. Artem Zvavitch , Co-Chair, Doctoral Dissertation Committee Dr. Matthieu Fradelizi , Co-Chair Dr. Dmitry Ryabogin , Members, Doctoral Dissertation Committee Dr. Fedor Nazarov Dr. Feodor F. Dragan Dr. Jonathan Maletic Accepted by Dr. Andrew Tonge , Chair, Department of Mathematical Sciences Dr. James L. Blank , Dean, College of Arts and Sciences TABLE OF CONTENTS LIST OF FIGURES . vi ACKNOWLEDGEMENTS ........................................ vii NOTATION ................................................. viii I Introduction 1 1 This Thesis . .2 2 Preliminaries . .4 2.1 Convex Geometry . .4 2.2 Basic functions and their properties . .5 2.3 Classical theorems . .6 2.4 Polarity . .8 2.5 Volume Product . .8 II The Discrete Slicing Problem 11 3 The Geometry of Numbers . 12 3.1 Introduction . 12 3.2 Lattices . 12 3.3 Minkowski's Theorems . 14 3.4 The Ehrhart Polynomial . 15 4 Geometric Tomography . 17 4.1 Introduction . 17 iii 4.2 Projections of sets . 18 4.3 Sections of sets . 19 4.4 The Isomorphic Busemann-Petty Problem . 20 5 The Discrete Slicing Problem . 22 5.1 Introduction . 22 5.2 Solution in Z2 ............................................. 23 5.3 Approach via Discrete F. John Theorem . 23 5.4 Solution using known inequalities . 26 5.5 Solution for Unconditional Bodies . 27 5.6 Discrete Brunn's Theorem . -
Classification of Ehrhart Polynomials of Integral Simplices Akihiro Higashitani
Classification of Ehrhart polynomials of integral simplices Akihiro Higashitani To cite this version: Akihiro Higashitani. Classification of Ehrhart polynomials of integral simplices. 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), 2012, Nagoya, Japan. pp.587-594. hal-01283113 HAL Id: hal-01283113 https://hal.archives-ouvertes.fr/hal-01283113 Submitted on 5 Mar 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. FPSAC 2012, Nagoya, Japan DMTCS proc. AR, 2012, 587–594 Classification of Ehrhart polynomials of integral simplices Akihiro Higashitani y Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan Abstract. Let δ(P) = (δ0; δ1; : : : ; δd) be the δ-vector of an integral convex polytope P of dimension d. First, by Pd using two well-known inequalities on δ-vectors, we classify the possible δ-vectors with i=0 δi ≤ 3. Moreover, by Pd means of Hermite normal forms of square matrices, we also classify the possible δ-vectors with i=0 δi = 4. In Pd Pd addition, for i=0 δi ≥ 5, we characterize the δ-vectors of integral simplices when i=0 δi is prime. -
Unimodular Triangulations of Dilated 3-Polytopes
Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc. Tom 74 (2013), vyp. 2 2013, Pages 293–311 S 0077-1554(2014)00220-X Article electronically published on April 9, 2014 UNIMODULAR TRIANGULATIONS OF DILATED 3-POLYTOPES F. SANTOS AND G. M. ZIEGLER Abstract. A seminal result in the theory of toric varieties, by Knudsen, Mumford and Waterman (1973), asserts that for every lattice polytope P there is a positive integer k such that the dilated polytope kP has a unimodular triangulation. In dimension 3, Kantor and Sarkaria (2003) have shown that k = 4 works for every polytope. But this does not imply that every k>4 works as well. We here study the values of k for which the result holds, showing that: (1) It contains all composite numbers. (2) It is an additive semigroup. These two properties imply that the only values of k that may not work (besides 1 and 2, which are known not to work) are k ∈{3, 5, 7, 11}.Withanad-hocconstruc- tion we show that k =7andk = 11 also work, except in this case the triangulation cannot be guaranteed to be “standard” in the boundary. All in all, the only open cases are k =3andk =5. § 1. Introduction Let Λ ⊂ Rd be an affine lattice. A lattice polytope (or an integral polytope [4]) is a polytope P with all its vertices in Λ. We are particularly interested in lattice (full- dimensional) simplices. The vertices of a lattice simplex Δ are an affine basis for Rd, hence they induce a sublattice ΛΔ of Λ of index equal to the normalized volume of Δ with respect to Λ. -
Generating Combinatorial Complexes of Polyhedral Type
transactions of the american mathematical society Volume 309, Number 1, September 1988 GENERATING COMBINATORIAL COMPLEXES OF POLYHEDRAL TYPE EGON SCHULTE ABSTRACT. The paper describes a method for generating combinatorial com- plexes of polyhedral type. Building blocks B are implanted into the maximal simplices of a simplicial complex C, on which a group operates as a combinato- rial reflection group. Of particular interest is the case where B is a polyhedral block and C the barycentric subdivision of a regular incidence-polytope K together with the action of the automorphism group of K. 1. Introduction. In this paper we discuss a method for generating certain types of combinatorial complexes. We generalize ideas which were recently applied in [23] for the construction of tilings of the Euclidean 3-space E3 by dodecahedra. The 120-cell {5,3,3} in the Euclidean 4-space E4 is a regular convex polytope whose facets and vertex-figures are dodecahedra and 3-simplices, respectively (cf. Coxeter [6], Fejes Toth [12]). When the 120-cell is centrally projected from some vertex onto the 3-simplex T spanned by the neighboured vertices, then a dissection of T into dodecahedra arises (an example of a central projection is shown in Figure 1). This dissection can be used as a building block for a tiling of E3 by dodecahedra. In fact, in the barycentric subdivision of the regular tessellation {4,3,4} of E3 by cubes each 3-simplex can serve as as a fundamental region for the symmetry group W of {4,3,4} (cf. Coxeter [6]). Therefore, mapping T affinely onto any of these 3-simplices and applying all symmetries in W turns the dissection of T into a tiling T of the whole space E3. -
The $ H^* $-Polynomial of the Order Polytope of the Zig-Zag Poset
THE h∗-POLYNOMIAL OF THE ORDER POLYTOPE OF THE ZIG-ZAG POSET JANE IVY COONS AND SETH SULLIVANT Abstract. We describe a family of shellings for the canonical tri- angulation of the order polytope of the zig-zag poset. This gives a new combinatorial interpretation for the coefficients in the numer- ator of the Ehrhart series of this order polytope in terms of the swap statistic on alternating permutations. 1. Introduction and Preliminaries The zig-zag poset Zn on ground set {z1,...,zn} is the poset with exactly the cover relations z1 < z2 > z3 < z4 > . That is, this partial order satisfies z2i−1 < z2i and z2i > z2i+1 for all i between n−1 1 and ⌊ 2 ⌋. The order polytope of Zn, denoted O(Zn) is the set n of all n-tuples (x1,...,xn) ∈ R that satisfy 0 ≤ xi ≤ 1 for all i and xi ≤ xj whenever zi < zj in Zn. In this paper, we introduce the “swap” permutation statistic on alternating permutations to give a new combinatorial interpretation of the numerator of the Ehrhart series of O(Zn). We began studying this problem in relation to combinatorial proper- ties of the Cavender-Farris-Neyman model with a molecular clock (or CFN-MC model) from mathematical phylogenetics [4]. We were inter- ested in the polytope associated to the toric variety obtained by ap- plying the discrete Fourier transform to the Cavender-Farris-Neyman arXiv:1901.07443v2 [math.CO] 1 Apr 2020 model with a molecular clock on a given rooted binary phylogenetic tree. We call this polytope the CFN-MC polytope. -
Triangulations of Integral Polytopes, Examples and Problems
数理解析研究所講究録 955 巻 1996 年 133-144 133 Triangulations of integral polytopes, examples and problems Jean-Michel KANTOR We are interested in polytopes in real space of arbitrary dimension, having vertices with integral co- ordinates: integral polytopes. The recent increase of interest for the study of these $\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}$ topes and their triangulations has various motivations; let us mention the main ones: . the beautiful theory of toric varieties has built a bridge between algebraic geometry and the combina- torics of these integral polytopes [12]. Triangulations of cones and polytopes occur naturally for example in problems of existence of crepant resolution of singularities $[1,5]$ . The work of the school of $\mathrm{I}.\mathrm{M}$ . Gelfand on secondary polytopes gives a new insight on triangulations, with applications to algebraic geometry and group theory [13]. In statistical physics, random tilings lead to some interesting problems dealing with triangulations of order polytopes $[6,29]$ . With these motivations in mind, we introduce new tools: Generalizations of the Ehrhart polynomial (counting points “modulo congruence”), discrete length between integral points (and studying the geometry associated to it), arithmetic Euler-Poincar\'e formula which gives, in dimension 3, the Ehrhart polynomial in terms of the $\mathrm{f}$-vector of a minimal triangulation of the polytope (Theorem 7). Finally, let us mention the results in dimension 2 of the late Peter Greenberg, they led us to the study of “Arithmetical $\mathrm{P}\mathrm{L}$-topology” which, we believe with M. Gromov, D. Sullivan, and P. Vogel, has not yet revealed all its beauties. -
Frequently Asked Questions in Polyhedral Computation
Frequently Asked Questions in Polyhedral Computation http://www.ifor.math.ethz.ch/~fukuda/polyfaq/polyfaq.html Komei Fukuda Swiss Federal Institute of Technology Lausanne and Zurich, Switzerland [email protected] Version June 18, 2004 Contents 1 What is Polyhedral Computation FAQ? 2 2 Convex Polyhedron 3 2.1 What is convex polytope/polyhedron? . 3 2.2 What are the faces of a convex polytope/polyhedron? . 3 2.3 What is the face lattice of a convex polytope . 4 2.4 What is a dual of a convex polytope? . 4 2.5 What is simplex? . 4 2.6 What is cube/hypercube/cross polytope? . 5 2.7 What is simple/simplicial polytope? . 5 2.8 What is 0-1 polytope? . 5 2.9 What is the best upper bound of the numbers of k-dimensional faces of a d- polytope with n vertices? . 5 2.10 What is convex hull? What is the convex hull problem? . 6 2.11 What is the Minkowski-Weyl theorem for convex polyhedra? . 6 2.12 What is the vertex enumeration problem, and what is the facet enumeration problem? . 7 1 2.13 How can one enumerate all faces of a convex polyhedron? . 7 2.14 What computer models are appropriate for the polyhedral computation? . 8 2.15 How do we measure the complexity of a convex hull algorithm? . 8 2.16 How many facets does the average polytope with n vertices in Rd have? . 9 2.17 How many facets can a 0-1 polytope with n vertices in Rd have? . 10 2.18 How hard is it to verify that an H-polyhedron PH and a V-polyhedron PV are equal? . -
Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations
Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations Jean Gallier and Jocelyn Quaintance Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] April 20, 2017 2 3 Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations Jean Gallier Abstract: Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, com- puter vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended for a broad audience of mathematically inclined readers. One of my (selfish!) motivations in writing these notes was to understand the concept of shelling and how it is used to prove the famous Euler-Poincar´eformula (Poincar´e,1899) and the more recent Upper Bound Theorem (McMullen, 1970) for polytopes. Another of my motivations was to give a \correct" account of Delaunay triangulations and Voronoi diagrams in terms of (direct and inverse) stereographic projections onto a sphere and prove rigorously that the projective map that sends the (projective) sphere to the (projective) paraboloid works correctly, that is, maps the Delaunay triangulation and Voronoi diagram w.r.t. the lifting onto the sphere to the Delaunay diagram and Voronoi diagrams w.r.t. the traditional lifting onto the paraboloid. Here, the problem is that this map is only well defined (total) in projective space and we are forced to define the notion of convex polyhedron in projective space.