On Volume and Surface Area of Parallel Sets

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.

19 pages

Cited by in corpus (13)

• Dimensions, Whitney covers, and tubular neighborhoods
• Characterization of Minkowski measurability in terms of surface area
• Fractal curvature measures and Minkowski content for one-dimensional self-conformal sets
• Estimation of fractal dimension and fractal curvatures from digital images
• Fine properties of the curvature of arbitrary closed sets
• Minkowski measurability of infinite conformal graph directed systems and application to Apollonian packings
• Curvature-direction measures of self-similar sets
• Localization results for Minkowski contents
• Curvature Bounds for Neighborhoods of Self-Similar Sets
• On existence of minimizers for weighted $L^p$-Hardy inequalities on $C^{1,\gamma }$-domains with compact boundary
• Fractal zeta functions at infinity and the $\varphi$-shell Minkowski content
• Optimal Control of Continuity Equations
• Lower S-Dimension of Fractal Sets