Consider the collection of all the simplices spanned by some n-point set in R^d. There are several results showing that simplices defined in this way must overlap very much. In this talk I focus on the generalization of these results to 'curvy' simplices.

Specifically, Pach showed that every d+1 sets of points, Q₁, …, Q_d+1, in R^d contain linearly-sized subsets P_i in Q_i such that all the transversal simplices that they span intersect. In joint work with Alfredo Hubard, we show, by means of an example, that a topological extension of Pach's theorem does not hold with subsets of size C(log n)^1/(d-1). We show that this is tight in dimension 2, for all surfaces other than S². Surprisingly, the optimal bound for S² is (log n)^1/2. This improves upon results of Barany, Meshulam, Nevo, Tancer.