## Abstract

The following result was proved by Bárány in 1982: For every d≥1, there exists c_{d}>0 such that for every n-point set S in ℝ^{d}, there is a point p∈ℝ^{d} contained in at least c_{d}n^{d+1}-O(n^{d}) of the d-dimensional simplices spanned by S. We investigate the largest possible value of c_{d}. It was known that c_{d}≤1/(2^{d}(d+1)!) (this estimate actually holds for every point set S). We construct sets showing that c_{d}≤(d+1)^{-(d+1)}, and we conjecture that this estimate is tight. The best known lower bound, due to Wagner, is c_{d}≥γ_{d}:=(d^{2}+1)/((d+1)!(d+1)^{d+1}); in his method, p can be chosen as any centerpoint of S. We construct n-point sets with a centerpoint that is contained in no more than γ_{d}n^{d+1}+O(n^{d}) simplices spanned by S, thus showing that the approach using an arbitrary centerpoint cannot be further improved. We also prove that for every n-point set S⊂ℝ^{d}, there exists a (d-2)-flat that stabs at least c_{d,d-2}n^{3}-O(n^{2}) of the triangles spanned by S, with c_{d,d-2}≥1/24(1-1/(2d-1)^{2}). To this end, we establish an equipartition result of independent interest (generalizing planar results of Buck and Buck and of Ceder): Every mass distribution in ℝ^{d} can be divided into 4d-2 equal parts by 2d-1 hyperplanes intersecting in a common (d-2)-flat.

Original language | English |
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Pages (from-to) | 321-338 |

Number of pages | 18 |

Journal | Discrete and Computational Geometry |

Volume | 43 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2010 |

Externally published | Yes |

## Keywords

- Centerpoint
- Cohomological index
- Equipartition
- Equivariant map
- Selection lemma
- Simplex