On Volume and Surface Area of Parallel Sets

Details On Volume and Surface Area of Parallel Sets On Volume and Surface Area of Parallel Sets

2009-05-20

arxiv.org/abs/0905.3279v2

Indiana University Mathematics Journal 59, 5 (2010), 1661-1685

Mathematics

Classical Analysis and ODEs

19 pages

Scientific paper

10.1512/iumj.2010.59.4165

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.

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