Improved Bounds for Point Selections and Halving Hyperplanes in Higher Dimensions

Research output: Contribution to conferencePaperpeer-review

Abstract

Let (P, E) be a (d + 1)-uniform geometric hypergraph, where P is an n-point set in general position in Rd and E ⊆ !dP+1" is a collection of !!d+1n" d-dimensional simplices with vertices in P, for 0 < ! ≤ 1. We show that there is a point x ∈ Rd that pierces (equation presented) simplices in E, for any fixed δ > 0. This is a dramatic improvement in all dimensions d ≥ 3, over the previous lower bounds of the general form (equation presented), which date back to the seminal 1991 work of Alon, Bárány, Füredi and Kleitman. As a result, any n-point set in general position in Rd admits only (equation presented) halving hyperplanes, for any δ > 0, which is a significant improvement over the previously best known bound (equation presented) in all dimensions d ≥ 5. An essential ingredient of our proof is the following semi-algebraic Turán-type result of independent interest, which holds for any fixed δ > 0: Let (V1, ..., Vk, E) be a hypergraph of bounded semi-algebraic description complexity, whose vertices are in sufficiently general position in Rd. Suppose that |E| ≥ ε|V1|·...·|Vk| holds for some ε > 0, then there exist subsets (equation presented).

Original languageAmerican English
Pages4464-4501
Number of pages38
DOIs
StatePublished - 1 Jan 2024
Event35th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2024 - Alexandria, United States
Duration: 7 Jan 202410 Jan 2024

Conference

Conference35th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2024
Country/TerritoryUnited States
CityAlexandria
Period7/01/2410/01/24

All Science Journal Classification (ASJC) codes

  • Software
  • General Mathematics

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