Convex hulls of face-vertex incident vectors of 3-colorable polytopes

TL;DR

This paper investigates the convex hulls of face-vertex incident vectors in 3-face-colorable convex polytopes, revealing their structure and criteria for equivalence using Gale transforms.

Contribution

It introduces a detailed analysis of these convex hulls, including their vertex counts, combinatorial structures, and a criterion for their equivalence.

Findings

Convex hulls are d-polytopes with d+2 or d+3 vertices.

Gale transform and Gale diagram are used to determine combinatorial structure.

Provides a necessary and sufficient condition for convex hulls' combinatorial equivalence.

Abstract

The convex hulls of face-vertex incident vectors of 3-face-colorable convex polytopes are computed. It is found that every such convex hull is a $d$-polytope with $d+2$ or $d+3$ vertices. Utilizing Gale transform and Gale diagram, we calculate its combinatorial structure. Finally, a necessary and sufficient criterion for combinatorial equivalence of two such convex hulls is given.