Polytopes, Valuations, and the Euler Relation

1968 ◽  
Vol 20 ◽  
pp. 1412-1424 ◽  
Author(s):  
G. T. Sallee
Keyword(s):  
Convex Polytopes ◽  
Convex Polytope ◽  
Euler Relation ◽  
Dimensional Face

By a d-polytope we shall mean a d-dimensional convex polytope. We shall denote a j-dimensional face (or j-face) of a polytope by Fj. Every d-polytope P has proper j-faces for 0 ≦j ≦d — 1 and we shall also say that P is a d-face of itself. Observe that every face of a polytope is again a polytope. The collection of all convex polytopes shall be denoted by .

scholarly journals A short note on extreme points of certain polytopes

2020 ◽  
Vol 8 (1) ◽  
pp. 36-39
Author(s):  
Lei Cao ◽  
Ariana Hall ◽  
Selcuk Koyuncu
Keyword(s):  
Short Proof ◽  
Convex Polytopes ◽  
Short Note ◽  
Convex Polytope ◽  
Extreme Points ◽  
Alternative Proof ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

AbstractWe give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

Polytopes with an Axis Of Symmetry

1970 ◽  
Vol 22 (2) ◽  
pp. 265-287 ◽  
Author(s):  
P. McMullen ◽  
G. C. Shephard
Keyword(s):  
Symmetry Group ◽  
Linear Subspace ◽  
Convex Polytopes ◽  
Convex Polytope ◽  
Simple Manner ◽  
Gale Transform ◽  
Enumeration Problems ◽  
Axis Of Symmetry ◽  
Axes Of Symmetry ◽  
Gale Diagrams

During the last few years, Branko Grünbaum, Micha Perles, and others have made extensive use of Gale transforms and Gale diagrams in investigating the properties of convex polytopes. Up to the present, this technique has been applied almost entirely in connection with combinatorial and enumeration problems. In this paper we begin by showing that Gale transforms are also useful in investigating properties of an essentially metrical nature, namely the symmetries of a convex polytope. Our main result here (Theorem (10)) is that, in a manner that will be made precise later, the symmetry group of a polytope can be represented faithfully by the symmetry group of a Gale transform of its vertices. If a d-polytope P ⊂ Ed has an axis of symmetry A (that is, A is a linear subspace of Ed such that the reflection in A is a symmetry of P), then it is called axi-symmetric. Using Gale transforms we are able to determine, in a simple manner, the possible numbers and dimensions of axes of symmetry of axi-symmetric polytopes.

scholarly journals Computing Reeb dynamics on four-dimensional convex polytopes

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Julian Chaidez ◽  
Michael Hutchings
Keyword(s):  
Convex Polytopes ◽  
Convex Polytope

<p style='text-indent:20px;'>We study the combinatorial Reeb flow on the boundary of a four-dimensional convex polytope. We establish a correspondence between "combinatorial Reeb orbits" for a polytope, and ordinary Reeb orbits for a smoothing of the polytope, respecting action and Conley-Zehnder index. One can then use a computer to find all combinatorial Reeb orbits up to a given action and Conley-Zehnder index. We present some results of experiments testing Viterbo's conjecture and related conjectures. In particular, we have found some new examples of polytopes with systolic ratio <inline-formula><tex-math id="M1">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>.</p>

On the Number of Vertices of a Convex Polytope

1964 ◽  
Vol 16 ◽  
pp. 701-720 ◽  
Author(s):  
Victor Klee
Keyword(s):  
Sharp Estimate ◽  
Convex Polytopes ◽  
Convex Polytope ◽  
Linear Inequalities ◽  
Bounded Subset ◽  
Empty Interior ◽  
System Of Linear Inequalities ◽  
System Of Inequalities ◽  
Basic Solutions ◽  
Upper Estimate

As is well known, the theory of linear inequalities is closely related to the study of convex polytopes. If the bounded subset P of euclidean d-space has a non-empty interior and is determined by i linear inequalities in d variables, then P is a d-dimensional convex polytope (here called a d-polytope) which may have as many as i faces of dimension d — 1, and the vertices of this polytope are exactly the basic solutions of the system of inequalities. Thus, to obtain an upper estimate of the size of the computation problem which must be faced in solving a system of linear inequalities, it suffices to find an upper bound for the number f0(P) of vertices of a d-polytope P which has a given number fd-1(P) of (d — l)-faces. A weak bound of this sort was found by Saaty (14), and several authors have posed the problem of finding a sharp estimate.

Cell Complexes, Valuations, and the Euler Relation

1970 ◽  
Vol 22 (2) ◽  
pp. 235-241 ◽  
Author(s):  
M. A. Perles ◽  
G. T. Sallee
Keyword(s):  
Euclidean Space ◽  
Hausdorff Metric ◽  
Dimensional Euclidean Space ◽  
Cell Complexes ◽  
Finite Dimensional ◽  
Finite Set ◽  
Euler Relation ◽  
Set Of Points ◽  
The Relationship ◽  
Dimensional Face

1. Recently a number of functions have been shown to satisfy relations on polytopes similar to the classic Euler relation. Much of this work has been done by Shephard, and an excellent summary of results of this type may be found in [11]. For such functions, only continuity (with respect to the Hausdorff metric) is required to assure that it is a valuation, and the relationship between these two concepts was explored in [8]. It is our aim in this paper to extend the results obtained there to illustrate the relationship between valuations and the Euler relation on cell complexes.To fix our notions, we will suppose that everything takes place in a given finite-dimensional Euclidean space X.A polytope is the convex hull of a finite set of points and will be referred to as a d-polytope if it has dimension d. Polytopes have faces of all dimensions from 0 to d – 1 and each of these is in turn a polytope. A k-dimensional face will be termed simply a k-face.

Estimate Volumes of Convex Polytopes: A New Method Based on Particle Swarm Optimization

2014 ◽  
Vol 1022 ◽  
pp. 357-360
Author(s):  
Sheng Xu ◽  
Dun Bo Cai
Keyword(s):  
Particle Swarm Optimization ◽  
Convex Polytopes ◽  
Particle Swarm ◽  
Convex Polytope ◽  
New Method ◽  
Monte Carlo Algorithm ◽  
Swarm Optimization ◽  
Uniform Sampling ◽  
Sampling Process ◽  
Sample Points

The volume of a convex polytope is important for many applications, and generally #P-hard to compute. In many scenarios, an approximate value of the volume is sufficed to utilize. Existing methods for estimating the volumes were mostly based on the Monte Carlo algorithm or its variants, which required a near uniform sampling process with a large number of sample points. In this paper, we propose a new method to estimate the volumes of convex polytopes that needs relative less sample points. Our method firstly searches for fringe points that are inside and near the border of a convex polytope, by a new way of utilizing the particle swarm optimization (PSO) technique. Then, the set of fringe points is input to a tool called Qhull whose output value serves as an estimate of the real volume. Experimental results show that our method is efficient and gives result with high accuracy for many instances of 10-dimension and beyond.

A Remark on Convex Polytopes

1965 ◽  
Vol 8 (6) ◽  
pp. 829-830
Author(s):  
A. S. Glass
Keyword(s):  
Dimensional Space ◽  
Convex Polytopes ◽  
Convex Polytope ◽  
Finite Family ◽  
Alternative Proof

In this note we wish to present an alternative proof for the following well-known theorem [1, Theorem 16]: every convex polytope X in Euclidean n-dimensional space Rn is the intersection of a finite family of closed half-spaces. It will be supposed that the converse of this theorem has been verified by conventional arguments, namely: every bounded intersection of a finite family of closed half-spaces in Rn is a convex polytope [cf. 1, Theorem 15].

Maximum-Entropy Representations in Convex Polytopes: Applications to Spatial Interaction

1989 ◽  
Vol 21 (11) ◽  
pp. 1541-1546 ◽  
Author(s):  
P B Slater
Keyword(s):  
Maximum Entropy ◽  
Convex Combination ◽  
Spatial Interaction ◽  
Stochastic Matrix ◽  
Convex Polytopes ◽  
Convex Polytope ◽  
Extreme Points ◽  
Doubly Stochastic ◽  
Interaction Modeling

Of all representations of a given point situated in a convex polytope, as a convex combination of extreme points, there exists one for which the probability or weighting distribution has maximum entropy. The determination of this multiplicative or exponential distribution can be accomplished by inverting a certain bijection—developed by Rothaus and by Bregman—of convex polytopes into themselves. An iterative algorithm is available for this procedure. The doubly stochastic matrix with a given set of transversals (generalized diagonal products) can be found by means of this method. Applications are discussed of the Rothaus -Bregman map to a proof of Birkhoff's theorem and to the calculation of trajectories of points leading to stationary or equilibrium values of the generalized permanent, in particular in spatial interaction modeling.

scholarly journals The infimum of the volumes of convex polytopes of any given facet areas is 0

2014 ◽  
Vol 51 (4) ◽  
pp. 466-519 ◽  
Author(s):  
N. Abrosimov ◽  
E. Makai ◽  
A. Mednykh ◽  
Yu. Nikonorov ◽  
G. Rote
Keyword(s):  
Necessary Conditions ◽  
Convex Polytopes ◽  
Sufficient Conditions ◽  
Convex Polytope

We prove the theorem mentioned in the title for ℝnwheren≧ 3. The case of the simplex was known previously. Also the casen= 2 was settled, but there the infimum was some well-defined function of the side lengths. We also consider the cases of spherical and hyperbolicn-spaces. There we give some necessary conditions for the existence of a convex polytope with given facet areas and some partial results about sufficient conditions for the existence of (convex) tetrahedra.

An Elementary Proof of Gram's Theorem for Convex Polytopes

1967 ◽  
Vol 19 ◽  
pp. 1214-1217 ◽  
Author(s):  
G. C. Shephard
Keyword(s):  
Euclidean Space ◽  
Elementary Proof ◽  
Convex Polytopes ◽  
Convex Polytope ◽  
Interior Angle ◽  
Negative Number ◽  
The Face

Let P be a d-polytope (that is, a d-dimensional convex polytope in Euclidean space) and for 0 ≤ j ≤ d – 1 let (i = 1, . . . ,ƒj(P)) represent its j-faces. Associated with each face is a non-negative number ϕ(P, ), to be defined later, which is called the interior angle of P at the face .

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