Neighborly inscribed polytopes and Delaunay triangulations

TL;DR

This paper constructs a vast family of neighborly polytopes inscribed on smooth convex bodies, significantly advancing the understanding of inscribable polytopes and their combinatorial diversity, with implications for Delaunay triangulations.

Contribution

It introduces the first examples of inscribable neighborly polytopes beyond cyclic polytopes, establishing a superexponential lower bound on their combinatorial types.

Findings

Superexponentially many inscribable neighborly polytopes exist.

First examples of inscribable neighborly polytopes not cyclic.

Superexponential lower bound for neighborly Delaunay triangulations.

Abstract

We construct a large family of neighborly polytopes that can be realized with all the vertices on the boundary of any smooth strictly convex body. In particular, we show that there are superexponentially many combinatorially distinct neighborly polytopes that admit realizations inscribed on the sphere. These are the first examples of inscribable neighborly polytopes that are not cyclic polytopes, and provide the current best lower bound for the number of combinatorial types of inscribable polytopes (which coincides with the current best lower bound for the number of combinatorial types of polytopes). Via stereographic projections, this translates into a superexponential lower bound for the number of combinatorial types of (neighborly) Delaunay triangulations.