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.

On The Number of Faces of a Convex Polytope

1964 ◽  
Vol 16 ◽  
pp. 12-17 ◽  
Author(s):  
David Gale
Keyword(s):  
Simplex Method ◽  
Linear Equations ◽  
Convex Polytope ◽  
Linear Inequalities ◽  
Basic Solutions ◽  
Systems Of Linear Inequalities ◽  
Number Of Faces ◽  
Number Of Iterations

The following problem is as yet unsolved: Given a convex polytope with N vertices in n-space, what is the maximum number of (n — 1)-faces which it can have? Aside from its geometric interest this question arises in connection with solving systems of linear inequalities and linear equations in non-negative variables. The problem is equivalent to asking for the best bound on the number of basic solutions for such problems and hence a bound (though a weak one) for the number of iterations needed in the simplex method for solving linear programmes.

scholarly journals PC-solutions and quasi-solutions of the interval system of linear algebraic equations

2021 ◽  
Vol 17 (3) ◽  
pp. 262-276
Author(s):  
Sergei I. Noskov ◽  
◽  
Anatoly V. Lakeyev ◽  
Keyword(s):  
Interval Analysis ◽  
Linear Inequalities ◽  
Algebraic Equations ◽  
System Of Linear Inequalities ◽  
Interval System ◽  
Linear Algebraic Equations ◽  
System Of Inequalities ◽  
Nonlinear Condition ◽  
Left And Right ◽  
Intensive Development

The problem of solving the interval system of linear algebraic equations (ISLAEs) is one of the well-known problems of interval analysis, which is currently undergoing intensive development. In general, this solution represents a set, which may be given differently, de- pending on which quantifiers are related to the elements of the left and right sides of this system. Each set of solutions of ISLAE to be determined is described by the domain of compatibility of the corresponding system of linear inequalities and, normally, one nonlinear condition of the type of complementarity. It is difficult to work with them when solving specific problems. Therefore, in the case of nonemptiness in the process of solving the problem it is recommended to find a so-called PC-solution, based on the application of the technique known in the theory of multi-criterial choice, that presumes maximization of the solving capacity of the system of inequalities. If this set is empty, it is recommended to find a quasi- solution of ISLAE. The authors compare the approach proposed for finding PC- and/or quasi-solutions to the approach proposed by S. P. Shary, which is based on the application of the recognizing functional.

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.

Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities

2020 ◽  
Vol 52 (4) ◽  
pp. 1249-1283
Author(s):  
Masatoshi Kimura ◽  
Tetsuya Takine
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Continuous Time ◽  
State Transition ◽  
Convex Polytope ◽  
Linear Inequalities ◽  
Systems Of Linear Inequalities ◽  
Free Sets ◽  
Practical Implications

AbstractThis paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

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 Greedy systems of linear inequalities and lexicographically optimal solutions

2019 ◽  
Vol 53 (5) ◽  
pp. 1929-1935
Author(s):  
Satoru Fujishige
Keyword(s):  
Convex Functions ◽  
Linear Inequalities ◽  
Convex Polyhedra ◽  
Optimal Solutions ◽  
Integral Points ◽  
System Of Linear Inequalities ◽  
Discrete Convexity ◽  
The Greedy Algorithm

The present note reveals the role of the concept of greedy system of linear inequalities played in connection with lexicographically optimal solutions on convex polyhedra and discrete convexity. The lexicographically optimal solutions on convex polyhedra represented by a greedy system of linear inequalities can be obtained by a greedy procedure, a special form of which is the greedy algorithm of J. Edmonds for polymatroids. We also examine when the lexicographically optimal solutions become integral. By means of the Fourier–Motzkin elimination Murota and Tamura have recently shown the existence of integral points in a polyhedron arising as a subdifferential of an integer-valued, integrally convex function due to Favati and Tardella [Murota and Tamura, Integrality of subgradients and biconjugates of integrally convex functions. Preprint arXiv:1806.00992v1 (2018)], which can be explained by our present result. A characterization of integrally convex functions is also given.

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