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Discrete & Computational Geometry

, Volume 43, Issue 2, pp 321–338 | Cite as

Stabbing Simplices by Points and Flats

  • Boris Bukh
  • Jiří Matoušek
  • Gabriel Nivasch
Article

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+1O(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 3O(n 2) of the triangles spanned by S, with \(c_{d,d-2}\ge\frac{1}{24}(1-1/(2d-1)^{2})\) . 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.

Keywords

Centerpoint Equipartition Equivariant map Selection lemma Simplex Cohomological index 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsFine HallPrincetonUSA
  2. 2.Department of Applied Mathematics and Institute of Theoretical Computer Science (ITI)Charles UniversityPrague 1Czech Republic
  3. 3.Blavatnik School of Computer ScienceTel Aviv UniversityTel AvivIsrael

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