Title: On volume and surface area of parallel sets

Authors: Jan Rataj and Steffen Winter

Issue: Volume 59 (2010), Issue 5, 1661-1686

Abstract: The $r$-parallel set to a set $A$ in a Euclidean space consists of all points with distance at most $r$ from $A$. We clarify the relation between the volume and the surface area of parallel sets and study the asymptotic behaviour of both quantities as $r$ tends to $0$. We show, for instance, that in general, the existence of a (suitably rescaled) limit of the surface area implies the existence of the corresponding limit for the volume, known as the Minkowski content. A full characterisation is obtained for the case of self-similar fractal sets. Applications to stationary random sets are discussed as well, in particular, to the trajectory of the Brownian motion.