Measure-Theoretic Properties of Level Sets of Distance Functions

Abstract

We consider the level sets of distance functions from the point of view of geometric measure theory. This lays the foundation for further research that can be applied, among other uses, to the derivation of a shape calculus based on the level-set method. Particular focus is put on the $(n-1)$-dimensional Hausdorff measure of these level sets. We show that, starting from a bounded set, all sub-level sets of its distance function have finite perimeter. Furthermore, if a uniform-density condition is satisfied for the initial set, one can even show an upper bound for the perimeter that is uniform for all level sets. Our results are similar to existing results in the literature, with the important distinction that they hold for all level sets and not just almost all. We also present an example demonstrating that our results are sharp in the sense that no uniform upper bound can exist if our uniform-density condition is not satisfied. This is even true if the initial set is otherwise very regular (i.e., a bounded Caccioppoli set with smooth boundary).