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Considering $ntimes ntimes n$ stochastic tensors $(a_{ijk})$ (i.e., nonnegative hypermatrices in which every sum over one index $i$, $j$, or $k$, is 1), we study the polytope ($Omega_{n}$) of all these tensors, the convex set ($L_n$) of all tensors in $Omega_{n}$ with some positive diagonals, and the polytope ($Delta_n$) generated by the permutation tensors. We show that $L_n$ is almost the same as $Omega_{n}$ except for some boundary points. We also present an upper bound for the number of vertices of $Omega_{n}$.
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This paper is concerned with the extreme points of the polytopes of stochastic tensors. By a tensor we mean a multi-dimensional array over the real number field. A line-stochastic tensor is a nonnegative tensor in which the sum of all entries on each line (i.e., one free index) is equal to 1; a plane-stochastic tensor is a nonnegative tensor in which the sum of all entries on each plane (i.e., two free indices) is equal to 1. In enumerating extreme points of the polytopes of line- and plane-stochastic tensors of order 3 and dimension $n$, we consider the approach by linear optimization and present new lower and upper bounds. We also study the coefficient matrices that define the polytopes.
By a tensor we mean a multidimensional array (matrix) or hypermatrix over a number field. This article aims to set an account of the studies on the permanent functions of tensors. We formulate the definitions of 1-permanent, 2-permanent, and $k$-permanent of a tensor in terms of hyperplanes, planes and $k$-planes of the tensor; we discuss the polytopes of stochastic tensors; at end we present an extension of the generalized matrix function for tensors.
A question related to some conjectures of Lutwak about the affine quermassintegrals of a convex body $K$ in ${mathbb R}^n$ asks whether for every convex body $K$ in ${mathbb R}^n$ and all $1leqslant kleqslant n$ $$Phi_{[k]}(K):={rm vol}_n(K)^{-frac{1}{n}}left (int_{G_{n,k}}{rm vol}_k(P_F(K))^{-n},d u_{n,k}(F)right )^{-frac{1}{kn}}leqslant csqrt{n/k},$$ where $c>0$ is an absolute constant. We provide an affirmative answer for some broad classes of random polytopes. We also discuss upper bounds for $Phi_{[k]}(K)$ when $K=B_1^n$, the unit ball of $ell_1^n$, and explain how this special instance has implications for the case of a general unconditional convex body $K$.
A neighborliness property of marginal polytopes of hierarchical models, depending on the cardinality of the smallest non-face of the underlying simplicial complex, is shown. The case of binary variables is studied explicitly, then the general case is reduced to the binary case. A Markov basis for binary hierarchical models whose simplicial complexes is the complement of an interval is given.
It is known that any tropical polytope is the image under the valuation map of ordinary polytopes over the Puiseux series field. The latter polytopes are called lifts of the tropical polytope. We prove that any pure tropical polytope is the intersection of the tropical half-spaces given by the images under the valuation map of the facet-defining half-spaces of a certain lift. We construct this lift explicitly, taking into account geometric properties of the given polytope. Moreover, when the generators of the tropical polytope are in general position, we prove that the above property is satisfied for any lift. This solves a conjecture of Develin and Yu.
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