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

Abstract

We give lower and upper bounds on the constant in Pach's selection theorem, which says that for every (d+1)-colored set of points in R^d, with n points of each color, one can select cn points from each color so that the intersection of all the rainbow simplices with vertices in the selected points is nonempty. 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.

Keywords

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