Projective Equivalences of k-neighbourly Polytopes

TL;DR

This paper investigates the conditions under which point sets in Euclidean space can be projectively transformed into vertices of k-neighborly polytopes, establishing bounds related to McMullen's problem.

Contribution

It provides new bounds on the maximum size of point sets that can be transformed into k-neighborly polytopes, advancing understanding of projective equivalences of polytopes.

Findings

Established lower bound: d + ⌈d/k⌉ + 1

Established upper bound: 2d - k + 1

Connected results to McMullen's problem

Abstract

We prove the following theorem, which is related to McMullen's problem on projective transformations of polytopes; let $2\leq k\leq \lfloor{\frac{d}{2}}\rfloor$ and $\nu{(d, k)}$ be the largest number such that any set of $\nu{(d,k)}$ points lying in general position in $\mathbb{R}^d$ can be mapped by a permissible projective transformation onto the vertices of a k-neighborly polytope, then $d + \left\lceil{\frac{d}{k}}\right\rceil +1 \leq \nu{(d, k)} < 2d-k +1$.