Simplex volumes in hyperplane 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.