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arXiv · 2512.12757

Simplex volumes in hyperplane arrangements

Koki Furukawa
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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.

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Koki Furukawa. 2025-12-14. Simplex volumes in hyperplane arrangements. https://arxiv.org/abs/2512.12757

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