Distance functions, critical points, and the topology of random Čech complexes
Omer Bobrowski, Robert J. Adler · Homology Homotopy and Applications · 2014
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 [16,17] in which the Betti numbers of the random Čech complex based on P were studied.