Simplex volumes in hyperplane arrangements

2025/12/14 by Koki Furukawa, Furukawa, Koki
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #General position #Hyperplane #Limits and Structures in Graph Theory #Point processes and geometric inequalities #Position (finance) #Simplex #Unit (ring theory) #Volume (thermodynamics) #math.CO

paper · pdf · doi:10.48550/arxiv.2512.12757

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2025/12/14 · openalex created_date 2025/12/17 · openalex updated_date 2026/07/28

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 ℝ^d, (2) the maximum number of minimum/maximum d-volume d-simplices determined by an arrangement of n hyperplanes in ℝ^d, and (3) the maximum number D_d(n) such that any arrangement of n hyperplanes in ℝ^d in general position contains D_d(n) hyperplanes forming d-simplices of distinct d-volumes.

Citations

• Forbidden paths and cycles in ordered graphs and matrices

Related

• On an extremal problem connected with simplices
• An extremal problem of regular simplices-The higher dimensional case
• On the volume of hyperplane sections of a d-cube
• Erdős's unit distance problem and rigidity
• Unit Distance Problems