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Vertices of the polytope of polystochastic matrices and product constructions

Anna A. Taranenko

Abstract

A multidimensional nonnegative matrix is called polystochastic if the sum of its entries at each line is equal to $1$. The set of all polystochastic matrices of order $n$ and dimension $d$ is a convex polytope $Ω_n^d$. In the present paper, we compare known bounds on the number $V(n,d)$ of vertices of the polytope $Ω_n^d$, propose two constructions of vertices of $Ω_n^d$ based on multidimensional matrix multiplication, and list all vertices of the polytope $Ω_3^4$.

Vertices of the polytope of polystochastic matrices and product constructions

Abstract

A multidimensional nonnegative matrix is called polystochastic if the sum of its entries at each line is equal to . The set of all polystochastic matrices of order and dimension is a convex polytope . In the present paper, we compare known bounds on the number of vertices of the polytope , propose two constructions of vertices of based on multidimensional matrix multiplication, and list all vertices of the polytope .
Paper Structure (4 sections, 14 theorems, 33 equations)

This paper contains 4 sections, 14 theorems, 33 equations.

Table of Contents

  1. Definitions
  2. Preliminary results
  3. Bounds on the number of vertices
  4. Constructions of vertices by matrix products

Key Result

Proposition 1

The polytope $\Omega_n^d$ is a $(n-1)^d$-dimensional polytope in $\mathbb{R}^{n^d}$ with $n^d$ facets.

Theorems & Definitions (19)

  • Proposition 1
  • Proposition 2: JurRys.stochmatr
  • Proposition 3: JurRys.stochmatr
  • Proposition 4: JurRys.stochmatr
  • Proposition 5
  • Proposition 6: see, e.g., Bron.covexpoly
  • Proposition 7
  • Theorem 1: Keevash.existdesII, LinLur.hdimper
  • Proposition 8
  • Theorem 2
  • ...and 9 more