[Paper Review] Realizability and inscribability for some simplicial spheres and matroid polytopes

Moritz Firsching|arXiv (Cornell University)|Aug 11, 2015

Advanced Combinatorial Mathematics41 references3 citations

TL;DR

This paper applies non-linear optimization techniques to realize and inscribe matroid polytopes and simplicial spheres, achieving a complete classification of neighborly 4-, 6-, and 7-dimensional polytopes with 11 vertices, neighborly 5-polytopes with 10 vertices, and simplicial 3-spheres with 10 vertices into polytopal and non-polytopal types, with many realizable polytopes also found to be inscribable.

ABSTRACT

We show that non-linear optimization techniques can successfully be applied to realize and to inscribe matroid polytopes and simplicial spheres. Thus we obtain a variety of results, which include 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.

Motivation & Objective

• To determine which simplicial spheres and matroid polytopes can be realized as convex polytopes using non-linear optimization.
• To classify neighborly polytopes of specific dimensions and vertex counts, particularly in dimensions 4, 5, 6, and 7.
• To distinguish between polytopal and non-polytopal simplicial 3-spheres with 10 vertices.
• To investigate the inscribability of realizable polytopes, especially those arising from neighborly configurations.

Proposed method

• Employing non-linear optimization to solve the realizability problem for matroid polytopes and simplicial spheres.
• Using optimization to test whether a given combinatorial sphere or matroid polytope can be embedded as a convex polytope in Euclidean space.
• Applying the same framework to test inscribability, i.e., whether a polytope can be realized with all vertices on a sphere.
• Leveraging computational algebraic geometry and numerical continuation methods to handle the non-linear constraints of realization and inscribability.
• Systematically analyzing neighborly configurations in low dimensions to achieve complete classifications.
• Validating results through computational verification of realizability and inscribability conditions.

Experimental results

Research questions

• RQ1 Which simplicial 3-spheres with 10 vertices are polytopal, and which are not?
• RQ2 What is the complete classification of neighborly 5-polytopes with 10 vertices?
• RQ3 How many neighborly 4-, 6-, and 7-dimensional polytopes with 11 vertices exist, and which are realizable?
• RQ4 To what extent are the realizable neighborly polytopes also inscribable?
• RQ5 Can non-linear optimization effectively resolve realizability and inscribability for matroid polytopes and simplicial spheres?

Key findings

• A complete classification of neighborly 4-, 6-, and 7-dimensional polytopes with 11 vertices was achieved using non-linear optimization.