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


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 rainbow d-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.

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    Although we are interested in the dependence of \(c^{\sup }_d\) on d, we call it a constant emphasizing its independence on the size of the sets \(X_i\).


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We thank Erik Aas for participating at the initial stage of this project. We also thank Karim Adiprasito for fruitful discussions about the minimum solid angle in a simplex and Andrew Suk for a brief discussion on the expected improved lower bound by Fox, Pach, and Suk. R. K. was supported by the Russian Foundation for Basic Research Grant 15-31-20403 (mol_a_ved) and grant 15-01-99563. J. K., Z. P., and M. T. were partially supported by ERC Advanced Research Grant No. 267165 (DISCONV) and by the project CE-ITI (GAČR P202/12/G061) of the Czech Science Foundation. J. K. was also partially supported by Swiss National Science Foundation Grants 200021-137574 and 200020-14453. P. P., Z. P., and M. T. were partially supported by the Charles University Grant GAUK 421511. P. P. was also partially supported by the Charles University Grant SVV-2014-260107. Z. P. was also partially supported by the Charles University Grant SVV-2014-260103. Part of this work was done when M. T. was affiliated with Institutionen för matematik, Kungliga Tekniska Högskolan, 100 44 Stockholm.

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Correspondence to Jan Kynčl.

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Editor in Charge: János Pach

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Karasev, R., Kynčl, J., Paták, P. et al. Bounds for Pach’s Selection Theorem and for the Minimum Solid Angle in a Simplex. Discrete Comput Geom 54, 610–636 (2015).

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

Mathematics Subject Classification

  • 52C35
  • 52C10
  • 28A75
  • 28A33