Abstract:We study the dual variants of the Erdős'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.
From: Koki Furukawa [view email]
[v1]
Sun, 14 Dec 2025 16:43:49 UTC (29 KB)
[v2]
Tue, 10 Mar 2026 10:54:22 UTC (27 KB)
[v3]
Tue, 21 Jul 2026 10:57:20 UTC (26 KB)
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