On the Number of Vertices in a Hyperplane Section of a Polytope

TL;DR

This paper investigates the maximum number of vertices in hyperplane sections of convex polytopes, introduces a new algorithm for finding all such sections, and analyzes the sequence of vertices across different slices, supported by computational experiments.

Contribution

It provides tight bounds on vertices in hyperplane slices of polytopes, introduces a novel algorithm for enumerating sections, and analyzes vertex sequences with new computational data.

Findings

Established tight bounds on maximum vertices in hyperplane sections.

Developed a new algorithm to find all hyperplane sections of polytopes.

Generated new computational data for hypercube sections.

Abstract

We study the slices or sections of a convex polytope by affine hyperplanes. We present results on two key problems: First, we provide tight bounds on the maximum number of vertices attainable by a hyperplane slice of $d$-polytope (a sort of upper bound theorem) and discuss a new algorithm to find all sections. Second, we investigate the sequence of numbers of vertices produced by the different slices over all possible hyperplanes and analyze the gaps that arise in that sequence. We study these sequences for three-dimensional polytopes and for hypercubes. Our results were obtained with the help of large computational experiments, and we report on new data generated for hypercubes.