## Abstract

Let (P, E) be a (d + 1)-uniform geometric hypergraph, where P is an n-point set in general position in R^{d} and E ⊆ ^{!}_{d}^{P}_{+1}^{"} is a collection of !^{!}_{d+1}^{n"} d-dimensional simplices with vertices in P, for 0 < ! ≤ 1. We show that there is a point x ∈ R^{d} 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 R^{d} 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, ..., V_{k}, E) be a hypergraph of bounded semi-algebraic description complexity, whose vertices are in sufficiently general position in R^{d}. Suppose that |E| ≥ ε|V1|·...·|V_{k}| holds for some ε > 0, then there exist subsets (equation presented).

Original language | English |
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Pages | 4464-4501 |

Number of pages | 38 |

DOIs | |

State | Published - 1 Jan 2024 |

Event | 35th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2024 - Alexandria, United States Duration: 7 Jan 2024 → 10 Jan 2024 |

### Conference

Conference | 35th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2024 |
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Country/Territory | United States |

City | Alexandria |

Period | 7/01/24 → 10/01/24 |

## ASJC Scopus subject areas

- Software
- General Mathematics