Polytopes with few vertices and few facets

TL;DR

This paper proves that the number of combinatorial types of d-polytopes with a fixed small number of vertices and facets is bounded independently of the dimension d.

Contribution

It establishes a bound on the number of combinatorial types of polytopes with few vertices and facets, independent of the dimension.

Findings

Number of combinatorial types is bounded independently of dimension d.

Bound applies to polytopes with d+1+α vertices and d+1+β facets.

Provides a new understanding of polytope complexity in high dimensions.

Abstract

In this note we prove that the number of combinatorial types of $d$-polytopes with $d+1+\alpha$ vertices and $d+1+\beta$ facets is bounded by a constant independent of $d$.