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For a finite set of points $P$ in $R^d$, the function $d_P: R^d \to R^+$measures Euclidean distance to the set $P$. We study the number of criticalpoints of $d_P$ when $P$ is a Poisson process. In particular, we study thelimit behavior of $N_k$ - the number of critical points of $d_P$ with Morseindex $k$ - as the density of points grows. We present explicit computationsfor the normalized, limiting, expectations and variances of the $N_k$, as wellas distributional limit theorems. We link these results to recent results inwhich the Betti numbers of the random \v{C}ech complex based on $P$ werestudied.

Distance Functions, Critical Points, and the Topology of Random \v{C}ech Complexes