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 .

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.

scholarly journals An elementary proof of Minkowski's second inequality

1969 ◽  
Vol 10 (1-2) ◽  
pp. 177-181 ◽  
Author(s):  
I. Danicic
Keyword(s):  
Euclidean Space ◽  
Elementary Proof ◽  
Content Volume ◽  
Lattice Points ◽  
The Body ◽  
Dimensional Euclidean Space ◽  
Open Convex ◽  
Successive Minima ◽  
Linearly Independent ◽  
Independent Lattice

Let K be an open convex domain in n-dimensional Euclidean space, symmetric about the origin O, and of finite Jordan content (volume) V. With K are associated n positive constants λ1, λ2,…,λn, the ‘successive minima of K’ and n linearly independent lattice points (points with integer coordinates) P1, P2, …, Pn (not necessarily unique) such that all lattice points in the body λ,K are linearly dependent on P1, P2, …, Pj-1. The points P1,…, Pj lie in λK provided that λ > λj. For j = 1 this means that λ1K contains no lattice point other than the origin. Obviously

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 An Euler-type volume identity

1999 ◽  
Vol 59 (3) ◽  
pp. 495-508
Author(s):  
Kevin Callahan ◽  
Kathy Hann
Keyword(s):  
Elementary Proof ◽  
Three Dimensional ◽  
Convex Polytope ◽  
Image Position

In this paper we present an elementary proof of a congruence by subtraction relation. In order to prove congruence by subtraction, we produce a dissection relating equal sub-polytopes. An immediate consequence of this relation is an Euler-type volume identity in ℝ3 which appeared in the Unsolved Problems section of the December 1996 MAA Monthly.This Euler-type volume identity relates the volumes of subsets of a polytope called wedges that correspond to its faces, edges, and vertices. A wedge consists of the inward normal chords of the polytope emanating from a face, vertex, or edge. This identity is stated in the theorem below.Euler Volume Theorem. For any three dimensional convex polytope PThis identity follows immediately from

scholarly journals On Constant Metric Dimension of Some Generalized Convex Polytopes

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Xuewu Zuo ◽  
Abid Ali ◽  
Gohar Ali ◽  
Muhammad Kamran Siddiqui ◽  
Muhammad Tariq Rahim ◽  
...  
Keyword(s):  
Image Processing ◽  
Euclidean Space ◽  
Global Positioning System ◽  
Metric Space ◽  
Network Theory ◽  
Convex Polytopes ◽  
Metric Dimension ◽  
Positioning System ◽  
The Family ◽  
Global Positioning

Metric dimension is the extraction of the affine dimension (obtained from Euclidean space E d ) to the arbitrary metric space. A family ℱ = G n of connected graphs with n ≥ 3 is a family of constant metric dimension if dim G = k (some constant) for all graphs in the family. Family ℱ has bounded metric dimension if dim G n ≤ M , for all graphs in ℱ . Metric dimension is used to locate the position in the Global Positioning System (GPS), optimization, network theory, and image processing. It is also used for the location of hospitals and other places in big cities to trace these places. In this paper, we analyzed the features and metric dimension of generalized convex polytopes and showed that this family belongs to the family of bounded metric dimension.

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 Immersion of Manifolds

1960 ◽  
Vol 12 ◽  
pp. 529-534 ◽  
Author(s):  
Hans Samelson
Keyword(s):  
Euclidean Space ◽  
Tangent Bundle ◽  
Normal Bundle ◽  
Vector Fields ◽  
Elementary Proof ◽  
Homology Class ◽  
Oriented Manifold ◽  
Normal Degree ◽  
Special Case

In (3) R. Lashof and S. Smale proved among other things the following theorem. If the compact oriented manifold M is immersed into the oriented manifold M', with dim M' ≥ dim M + 2, then the normal degree of the immersion is equal to the Euler-Poincaré characteristic x of M reduced module the characteristic x’ of M'. If M’ is not compact, x' is replaced by 0. “Manifold” always means C∞-manifold. An immersion is a differentiable (that is, C∞) map f whose differential df is non-singular throughout. The normal degree is defined in a certain fashion using the normal bundle of M in M', derived from f, and injecting it into the tangent bundle of M'It is our purpose to give an elementary proof, using vector fields, of this theorem, and at the same time to identify the homology class that represents the normal degree (Theorem I), and to give a proof, using the theory of Morse, for the special case M’ = Euclidean space (Theorem II).

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.

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.

Sign in / Sign up

Export Citation Format

Share Document