Many regular triangulations and many polytopes

TL;DR

The paper establishes that for fixed dimension greater than three, the number of distinct labeled combinatorial types of polytopes with many vertices grows super-exponentially, doubling previous lower bounds, and also analyzes regular triangulations of neighborly polytopes.

Contribution

It provides new lower bounds on the number of combinatorial types of polytopes and regular triangulations, significantly improving previous estimates.

Findings

Number of polytope types grows as (n!)^{d-2} for fixed d>3.

Certain neighborly polytopes have at least (n!)^{floor((d-1)/2)} regular triangulations.

Results double the previous best lower bounds on polytope types.

Abstract

We show that for fixed $d>3$ and $n$ growing to infinity there are at least $(n!)^{d-2 \pm o(1)}$ different labeled combinatorial types of $d$-polytopes with $n$ vertices. This is about the square of the previous best lower bounds. As an intermediate step, we show that certain neighborly polytopes (such as particular realizations of cyclic polytopes) have at least $(n!)^{ \lfloor(d-1)/2\rfloor \pm o(1)}$ regular triangulations.