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 (Formula presented.) such that whenever (Formula presented.) are n-element subsets of (Formula presented.), we can find a point (Formula presented.) and subsets (Formula presented.) for every i∈[d+1], each of size at least cdn, such that p belongs to all rainbowd-simplices determined by (Formula presented.) simplices with one vertex in each Yi. We show a super-exponentially decreasing upper bound (Formula presented.). The ideas used in the proof of the upper bound also help us to prove Pach’s theorem with (Formula presented.), 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 (Formula presented.). 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 2d separations in the original proof. We also provide a measure version of Pach’s theorem.