Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, June 10, 2014, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Zuzana Safernová (Charles University)

Bounds for Pach's selection theorem

We focus on the estimates on 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 X₁, … , X_d+1 are n-element subsets of R^d, then we can find a point p in R^d and subsets Y_i ⊂ X_i  for every i=1,…, d+1, each of size at least c_d n, such that p belongs to all rainbow d-simplices determined by Y₁, … , Y_d+1, that is, simplices with one vertex in each Y_i.

We will discuss a lower and upper bound for the constant c_d, more concretely, 2^{-2^{d^2 + O(d)}} ≤ c_d ≤ 0.9975^d. In order to show the upper bound, we prove the fact that the minimum solid angle of every d-simplex is exponentially small.

Joint work with Honza Kynčl, Pavel Paták and Martin Tancer.

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