We consider the problem of computing the squared volume of the largest j-simplex contained in an n-dimensional polytope presented by its vertices (a V -polytope). We show that the related decision problem is W [1]-complete, with respect to the parameter j. We also improve the constant inapproximability factor given in [Packer, 2004, Discrete Applied Mathematics, 134], by showing that there are ...

We describe a class of facets of the polytope of convex combinations of the collection of even n × n permutation matrices. As a consequence, we prove the conjecture of Brualdi and Liu [4] that the number of facets of the polytope is not bounded by a polynomial in n.

We prove the central limit theorem for the volume and the f -vector of the random polytope Pn and the Poisson random polytope Πn in a fixed convex polytope P ⊂ IR. Here Pn is the convex hull of n random points in P , and Πn is the convex hull of the intersection of a Poisson process X(n), of intensity n, with P . A general lower bound on the variance is also proved. ∗Supported by Hungarian Nati...

We consider two polytopes. The quadratic assignment polytope QAP(n) is the convex hull of the set of tensors x⊗x, x ∈ Pn, where Pn is the set of n×n permutation matrices. The second polytope is defined as follows. For every permutation of vertices of the complete graph Kn we consider appropriate (n 2 ) × (n 2 ) permutation matrix of the edges of Kn. The Young polytope P ((n − 2, 2)) is the conv...

We prove that the cd-index of a convex polytope satisfies a strong monotonicity property with respect to the cd-indices of any face and its link. As a consequence, we prove for d-dimensional polytopes a conjecture of Stanley that the cd-index is minimized on the d-dimensional simplex. Moreover, we prove the upper bound theorem for the cd-index, namely that the cd-index of any d-dimensional poly...

A complete set of N + 1 mutually unbiased bases (MUBs) forms a convex polytope in the N2 − 1 dimensional space of N × N Hermitian matrices of unit trace. As a geometrical object such a polytope exists for all values of N , while it is unknown whether it can be made to lie within the body of density matrices unless N = pk, where p is prime. We investigate the polytope in order to see if some val...

The polytope model combines a set of methods and approaches based on a mathematical model to define a new general framework for automatic loop parallelization. The parallelization of n nested loops in this model proceeds in three steps: first modeling the loop nest by a convex polytope, second applying an affine coordinate transformation i.e. finding a schedule " t " and an allocation " a " on ...

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? . ....

We study the vertices and facets of the polytopes of partitions of numbers. The partition polytope n P is the convex hull of the set of incidence vectors of all partitions 1 2 2 ... n n x x nx = + + +. We show that the sequence 1 2 n P P P can be treated as an embedded chain. Dynamics of behavior of the vertices of n P , as n increases, is established. Some sufficient and some necessary conditi...

We prove that an equilibrium of a nondegenerate bimatrix game has index +1 if and only if it can be made the unique equilibrium of an extended game with additional strategies of one player. The main tool is the “dual construction”. A simplicial polytope, dual to the common best-response polytope of one player, has its facets subdivided into best-response regions, so that equilibria are complete...