Critical Point Theory for the Distance Function on PL-Spaces

Abstract

We study the (typical) singularities of the distance function from a point in the ambient space to a polytopal complex in Euclidean space. By means of the Goresky-MacPherson splitting we generalize the existing linear theory to the quadratic distance function. Critical points occur on `selected' strata of the PL-space, where normal Morse Data determine the change of topology. On the homology level there exist indices with multiplicities, combined in a Morse vector. We focus mainly on PL-manifolds. We give bounds, depending on the dimension of the strata. In low dimensions we describe singularity types. At the end we discuss relations with Discrete Morse Theory.