Many equiprojective polytopes

TL;DR

This paper investigates the diversity of 3D equiprojective polytopes, establishing a super-polynomial lower bound on their number of combinatorial types as a function of the projection vertex count.

Contribution

It provides a new lower bound of at least $k^{3k/2+o(k)}$ on the number of combinatorial types of $k$-equiprojective polytopes, using novel constructions and combinatorial bounds.

Findings

Number of combinatorial types grows super-polynomially with k

Constructs equiprojective polytopes via Minkowski sums

Relates to Goodman--Pollack order type bounds

Abstract

A $3$-dimensional polytope $P$ is $k$-equiprojective when the projection of $P$ along any line that is not parallel to a facet of $P$ is a polygon with $k$ vertices. In 1968, Geoffrey Shephard asked for a description of all equiprojective polytopes. It has been shown recently that the number of combinatorial types of $k$-equiprojective polytopes is at least linear as a function of $k$. Here, it is shown that there are at least $k^{3k/2+o(k)}$ such combinatorial types as $k$ goes to infinity. This relies on the Goodman--Pollack lower bound on the number of order types and on new constructions of equiprojective polytopes via Minkowski sums.