Distance functions, critical points, and the topology of random Čech complexes

Abstract

For a finite set of points P in Rd, the function dP: ℝd → ℝ+ measures Euclidean distance to the set P. We study the number of critical points of dP when P is a Poisson process. In particular, we study the limit behavior of Nk-the number of critical points of dP with Morse index k-as the density of points grows. We present explicit computations for the normalized limiting expectations and variances of the Nk, as well as distributional limit theorems.We link these results to recent results in [16, 17] in which the Betti numbers of the random Čech complex based on P were studied.