Random Gale diagrams and neighborly polytopes in high dimensions

TL;DR

This paper explores the random generation of Gale diagrams to produce high-dimensional polytopes, demonstrating that these polytopes are highly neighborly with high probability under certain conditions.

Contribution

It introduces a novel probabilistic method for generating combinatorially isomorphic polytopes via Gale diagrams, revealing their neighborliness properties in high dimensions.

Findings

Generated polytopes exhibit strong neighborliness in high dimensions.

High probability of neighborliness under growth assumptions.

Method confirms and extends Gale's 1956 suggestion.

Abstract

Taking up a suggestion of David Gale from 1956, we generate sets of combinatorially isomorphic polytopes by choosing their Gale diagrams at random. We find that in high dimensions, and under suitable assumptions on the growth of the involved parameters, the obtained polytopes have strong neighborliness properties, with high probability.