On Volume and Surface Area of Parallel Sets

TL;DR

This paper investigates the relationship between volume and surface area of parallel sets in Euclidean space, analyzing their asymptotic behavior as the offset radius approaches zero, with applications to fractals and stochastic processes.

Contribution

It establishes a link between the limits of surface area and volume for parallel sets and characterizes this relationship for self-similar fractals.

Findings

Existence of surface area limit implies volume limit for parallel sets.

Full characterization of limits for self-similar fractals.

Applications to Brownian motion trajectories.

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.