On the number of vertices of projective polytopes

TL;DR

This paper investigates the maximum number of vertices of projectively transformed convex hulls of point configurations, providing bounds and exploring related combinatorial problems involving Radon partitions and hyperplane arrangements.

Contribution

It introduces new bounds on the number of vertices of projective polytopes and connects these to Radon partitions and hyperplane arrangements, partially answering existing open questions.

Findings

Derived upper bounds for vertices of projective polytopes

Established bounds on minimal Radon partitions

Provided partial answers to questions by Pach and Szegedy

Abstract

Let $X$ be a configuration of $n$ points in $\mathbb{R}^d$. What is the maximum number of vertices that $conv(T(X))$ can have among all the possible permissible projective transformations $T$? In this paper, we investigate this and connected questions. After presenting several upper bounds, we study a closely related problem (via Gale transforms) concerning the number of minimal Radon partitions of a set of points. We then present some bounds for this number that enable us to partially answer a question due to Pach and Szegedy. We also discuss another related problem concerning the size of topes in arrangements of hyperplanes.