Extremal spherical polytopes and Borsuk's conjecture

TL;DR

This paper explores the generation and analysis of extremal spherical polytopes using gradient flow methods, providing insights into their combinatorics and potential links to Borsuk's conjecture.

Contribution

It introduces a numerical approach to generate anti-self-polar polytopes and analyzes their critical points and combinatorial properties, relating them to Borsuk's conjecture.

Findings

Generated anti-self-polar polytopes via gradient flow.

Proved results on critical points of the diameter functional.

Discussed potential connections to Borsuk's conjecture.

Abstract

We generate anti-self-polar polytopes via a numerical implementation of the gradient flow induced by the diameter functional on the space of all finite subsets of the sphere, and prove related results on the critical points of the diameter functional as well as results about the combinatorics of such polytopes. We also discuss potential connections to Borsuk's conjecture.