  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 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
which the Betti numbers of the random \v{C}ech complex based on $P$ were
studied.
