Realizability and inscribability for simplicial polytopes via nonlinear optimization

TL;DR

This paper demonstrates that nonlinear optimization can effectively realize and inscribe certain classes of polytopes and spheres, leading to complete classifications in specific dimensions and vertex counts.

Contribution

It introduces a novel application of nonlinear optimization techniques to classify neighborly polytopes and simplicial spheres, including their realizability and inscribability.

Findings

Complete classification of neighborly polytopes in dimensions 4, 6, 7 with 11 vertices

Classification of neighborly 5-polytopes with 10 vertices

Distinction between polytopal and non-polytopal simplicial 3-spheres with 10 vertices

Abstract

We show that nonlinear optimization techniques can successfully be applied to realize and to inscribe matroid polytopes and simplicial spheres. Thus we obtain a complete classification of neighborly polytopes of dimension $4$, $6$ and $7$ with $11$ vertices, of neighborly $5$-polytopes with $10$ vertices, as well as a complete classification of simplicial $3$-spheres with $10$ vertices into polytopal and non-polytopal spheres. Surprisingly many of the realizable polytopes are also inscribable.