Discrete & Computational Geometry

, Volume 54, Issue 3, pp 610–636 | Cite as

Bounds for Pach’s Selection Theorem and for the Minimum Solid Angle in a Simplex

  • Roman Karasev
  • Jan Kynčl
  • Pavel Paták
  • Zuzana Patáková
  • Martin Tancer
Article

Abstract

We estimate the selection constant in the following geometric selection theorem by Pach: For every positive integer d, there is a constant \(c_d > 0\) such that whenever \(X_1, \ldots , X_{d+1}\) are n-element subsets of \(\mathbb {R}^d\), we can find a point \({\mathbf {p}}\in \mathbb {R}^d\) and subsets \(Y_i \subseteq X_i\) for every \(i \in [d+1]\), each of size at least \(c_d n\), such that \({\mathbf {p}}\) belongs to all rainbowd-simplices determined by \(Y_1, \ldots , Y_{d+1}\), i.e., simplices with one vertex in each \(Y_i\). We show a super-exponentially decreasing upper bound \(c_d\le e^{-(1/2-o(1))(d \ln d)}\). The ideas used in the proof of the upper bound also help us to prove Pach’s theorem with \(c_d \ge 2^{-2^{d^2 + O(d)}}\), which is a lower bound doubly exponentially decreasing in d (up to some polynomial in the exponent). For comparison, Pach’s original approach yields a triply exponentially decreasing lower bound. On the other hand, Fox, Pach, and Suk recently obtained a hypergraph density result implying a proof of Pach’s theorem with \(c_d \ge 2^{-O(d^2\log d)}\). In our construction for the upper bound, we use the fact that the minimum solid angle of every d-simplex is super-exponentially small. This fact was previously unknown and might be of independent interest. For the lower bound, we improve the ‘separation’ part of the argument by showing that in one of the key steps only \(d+1\) separations are necessary, compared to \(2^d\) separations in the original proof. We also provide a measure version of Pach’s theorem.

Keywords

Pach’s selection theorem d-Dimensional simplex Solid angle Borel probability measure Weak convergence of measures 

Mathematics Subject Classification

52C35 52C10 28A75 28A33 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Moscow Institute of Physics and TechnologyInstitutskiy per. 9DolgoprudnyRussia
  2. 2.Institute for Information Transmission ProblemsRASMoscowRussia
  3. 3.Department of Applied Mathematics and Institute for Theoretical Computer ScienceCharles UniversityPrague 1Czech Republic
  4. 4.Alfréd Rényi Institute of MathematicsBudapestHungary
  5. 5.Chair of Combinatorial Geometry, École Polytechnique Fédérale de LausanneEPFL-SB-MATHGEOM-DCGLausanneSwitzerland
  6. 6.Department of AlgebraCharles UniversityPrague 8Czech Republic
  7. 7.IST AustriaKlosterneuburgAustria

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