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arXiv · 0905.3279

On Volume and Surface Area of Parallel Sets

Jan Rataj·Steffen Winter
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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.

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Jan Rataj, Steffen Winter. 2010-10-11. On Volume and Surface Area of Parallel Sets. https://doi.org/10.1512/iumj.2010.59.4165

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