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

Abstract

For a finite set of points P in R d , the function d P : R d R + measures Euclidean distance to the set P. We study the number of critical points of d P when P is a Poisson process. In particular, we study the limit behavior of N k -the number of critical points of d P with Morse index k-as the density of points grows. We present explicit computations for the normalized limiting expectations and variances of the N k , as well as distributional limit theorems. We link these results to recent results in