Simplex volumes in hyperplane arrangements

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