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

Understand

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.