Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations
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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. -
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. -
Locally Solid Riesz Spaces with Applications to Economics / Charalambos D
http://dx.doi.org/10.1090/surv/105 alambos D. Alipr Lie University \ Burkinshaw na University-Purdue EDITORIAL COMMITTEE Jerry L. Bona Michael P. Loss Peter S. Landweber, Chair Tudor Stefan Ratiu J. T. Stafford 2000 Mathematics Subject Classification. Primary 46A40, 46B40, 47B60, 47B65, 91B50; Secondary 28A33. Selected excerpts in this Second Edition are reprinted with the permissions of Cambridge University Press, the Canadian Mathematical Bulletin, Elsevier Science/Academic Press, and the Illinois Journal of Mathematics. For additional information and updates on this book, visit www.ams.org/bookpages/surv-105 Library of Congress Cataloging-in-Publication Data Aliprantis, Charalambos D. Locally solid Riesz spaces with applications to economics / Charalambos D. Aliprantis, Owen Burkinshaw.—2nd ed. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 105) Rev. ed. of: Locally solid Riesz spaces. 1978. Includes bibliographical references and index. ISBN 0-8218-3408-8 (alk. paper) 1. Riesz spaces. 2. Economics, Mathematical. I. Burkinshaw, Owen. II. Aliprantis, Char alambos D. III. Locally solid Riesz spaces. IV. Title. V. Mathematical surveys and mono graphs ; no. 105. QA322 .A39 2003 bib'.73—dc22 2003057948 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
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. -
Techniques of Variational Analysis
J. M. Borwein and Q. J. Zhu Techniques of Variational Analysis An Introduction October 8, 2004 Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo To Tova, Naomi, Rachel and Judith. To Charles and Lilly. And in fond and respectful memory of Simon Fitzpatrick (1953-2004). Preface Variational arguments are classical techniques whose use can be traced back to the early development of the calculus of variations and further. Rooted in the physical principle of least action they have wide applications in diverse ¯elds. The discovery of modern variational principles and nonsmooth analysis further expand the range of applications of these techniques. The motivation to write this book came from a desire to share our pleasure in applying such variational techniques and promoting these powerful tools. Potential readers of this book will be researchers and graduate students who might bene¯t from using variational methods. The only broad prerequisite we anticipate is a working knowledge of un- dergraduate analysis and of the basic principles of functional analysis (e.g., those encountered in a typical introductory functional analysis course). We hope to attract researchers from diverse areas { who may fruitfully use varia- tional techniques { by providing them with a relatively systematical account of the principles of variational analysis. We also hope to give further insight to graduate students whose research already concentrates on variational analysis. Keeping these two di®erent reader groups in mind we arrange the material into relatively independent blocks. We discuss various forms of variational princi- ples early in Chapter 2. We then discuss applications of variational techniques in di®erent areas in Chapters 3{7. -
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 . -
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? . -
Convex Sets and Convex Functions 1 Convex Sets
Convex Sets and Convex Functions 1 Convex Sets, In this section, we introduce one of the most important ideas in economic modelling, in the theory of optimization and, indeed in much of modern analysis and computatyional mathematics: that of a convex set. Almost every situation we will meet will depend on this geometric idea. As an independent idea, the notion of convexity appeared at the end of the 19th century, particularly in the works of Minkowski who is supposed to have said: \Everything that is convex interests me." We discuss other ideas which stem from the basic definition, and in particular, the notion of a convex and concave functions which which are so prevalent in economic models. The geometric definition, as we will see, makes sense in any vector space. Since, for the most of our work we deal only with Rn, the definitions will be stated in that context. The interested student may, however, reformulate the definitions either, in an ab stract setting or in some concrete vector space as, for example, C([0; 1]; R)1. Intuitively if we think of R2 or R3, a convex set of vectors is a set that contains all the points of any line segment joining two points of the set (see the next figure). Here is the definition. Definition 1.1 Let u; v 2 V . Then the set of all convex combinations of u and v is the set of points fwλ 2 V : wλ = (1 − λ)u + λv; 0 ≤ λ ≤ 1g: (1.1) In, say, R2 or R3, this set is exactly the line segment joining the two points u and v. -
A Tutorial on Convex Optimization
A Tutorial on Convex Optimization Haitham Hindi Palo Alto Research Center (PARC), Palo Alto, California email: [email protected] Abstract— In recent years, convex optimization has be- quickly deduce the convexity (and hence tractabil- come a computational tool of central importance in engi- ity) of many problems, often by inspection; neering, thanks to it’s ability to solve very large, practical 2) to review and introduce some canonical opti- engineering problems reliably and efficiently. The goal of this tutorial is to give an overview of the basic concepts of mization problems, which can be used to model convex sets, functions and convex optimization problems, so problems and for which reliable optimization code that the reader can more readily recognize and formulate can be readily obtained; engineering problems using modern convex optimization. 3) to emphasize modeling and formulation; we do This tutorial coincides with the publication of the new book not discuss topics like duality or writing custom on convex optimization, by Boyd and Vandenberghe [7], who have made available a large amount of free course codes. material and links to freely available code. These can be We assume that the reader has a working knowledge of downloaded and used immediately by the audience both linear algebra and vector calculus, and some (minimal) for self-study and to solve real problems. exposure to optimization. Our presentation is quite informal. Rather than pro- I. INTRODUCTION vide details for all the facts and claims presented, our Convex optimization can be described as a fusion goal is instead to give the reader a flavor for what is of three disciplines: optimization [22], [20], [1], [3], possible with convex optimization. -
36 COMPUTATIONAL CONVEXITY Peter Gritzmann and Victor Klee
36 COMPUTATIONAL CONVEXITY Peter Gritzmann and Victor Klee INTRODUCTION The subject of Computational Convexity draws its methods from discrete mathe- matics and convex geometry, and many of its problems from operations research, computer science, data analysis, physics, material science, and other applied ar- eas. In essence, it is the study of the computational and algorithmic aspects of high-dimensional convex sets (especially polytopes), with a view to applying the knowledge gained to convex bodies that arise in other mathematical disciplines or in the mathematical modeling of problems from outside mathematics. The name Computational Convexity is of more recent origin, having first ap- peared in print in 1989. However, results that retrospectively belong to this area go back a long way. In particular, many of the basic ideas of Linear Programming have an essentially geometric character and fit very well into the conception of Compu- tational Convexity. The same is true of the subject of Polyhedral Combinatorics and of the Algorithmic Theory of Polytopes and Convex Bodies. The emphasis in Computational Convexity is on problems whose underlying structure is the convex geometry of normed vector spaces of finite but generally not restricted dimension, rather than of fixed dimension. This leads to closer con- nections with the optimization problems that arise in a wide variety of disciplines. Further, in the study of Computational Convexity, the underlying model of com- putation is mainly the binary (Turing machine) model that is common in studies of computational complexity. This requirement is imposed by prospective appli- cations, particularly in mathematical programming. For the study of algorithmic aspects of convex bodies that are not polytopes, the binary model is often aug- mented by additional devices called \oracles." Some cases of interest involve other models of computation, but the present discussion focuses on aspects of computa- tional convexity for which binary models seem most natural. -
OPTIMIZATION METHODS: CLASS 3 Linearity, Convexity, ANity
OPTIMIZATION METHODS: CLASS 3 Linearity, convexity, anity The exercises are on the opposite side. D: A set A ⊆ Rd is an ane space, if A is of the form L + v for some linear space L and a shift vector v 2 Rd. By A is of the form L + v we mean a bijection between vectors of L and vectors of A given as b(u) = u + v. Each ane space has a dimension, dened as the dimension of its associated linear space L. D: A vector is an ane combination of a nite set of vectors if Pn , where x a1; a2; : : : an x = i=1 αiai are real number satisfying Pn . αi i=1 αi = 1 A set of vectors V ⊆ Rd is anely independent if it holds that no vector v 2 V is an ane combination of the rest. D: GIven a set of vectors V ⊆ Rd, we can think of its ane span, which is a set of vectors A that are all possible ane combinations of any nite subset of V . Similar to the linear spaces, ane spaces have a nite basis, so we do not need to consider all nite subsets of V , but we can generate the ane span as ane combinations of the base. D: A hyperplane is any ane space in Rd of dimension d − 1. Thus, on a 2D plane, any line is a hyperplane. In the 3D space, any plane is a hyperplane, and so on. A hyperplane splits the space Rd into two halfspaces. We count the hyperplane itself as a part of both halfspaces.