Simplex volumes in hyperplane arrangements

TL;DR

This paper explores the combinatorial geometry of hyperplane arrangements by analyzing the maximum counts of simplices with specific volume properties and the diversity of simplex volumes in general position.

Contribution

It introduces dual variants of classic distance problems, focusing on simplex volumes in hyperplane arrangements and establishing bounds on their counts and volume diversity.

Findings

Bounded the maximum number of unit-volume simplices in hyperplane arrangements.

Determined the maximum number of min/max volume simplices in such arrangements.

Established the minimum number of distinct-volume simplices in general position arrangements.

Abstract

We study the dual variants of the Erd\H{o}s's distinct distances and unit distance problems. Instead of considering distances determined by points, we consider simplex volumes determined by hyperplanes. We investigate: (1) the maximum number of unit $d$-volume $d$-simplices determined by an arrangement of $n$ hyperplanes in $\mathbb{R}^d$, (2) the maximum number of minimum/maximum $d$-volume $d$-simplices determined by an arrangement of $n$ hyperplanes in $\mathbb{R}^d$, and (3) the maximum number $D_d(n)$ such that any arrangement of $n$ hyperplanes in $\mathbb{R}^d$ in general position contains $D_d(n)$ hyperplanes forming $d$-simplices of distinct $d$-volumes.