On volume and surface area of parallel sets
By Jan Rataj, Steffen Winter. 2009
59 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/e7ad7670296b27ae26d04b73d208f5e504675306
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In Applied Mathematics
In Mathematical Physics
Citations in the corpus
- Normal approximation for stabilizing functionals (2017, unknown): “6 in [33], the hypotheses yield r0, C1 ∈ (0,∞) such that H(∂Kr) ≤ C1, r ∈ (0, r0).”
- Optimal control of continuity equations (2015, background): “Such sets have nice regularity properties; in particular, ∂A is (n− 1)-rectifiable [21] and σ(∂A) ≤ nαn(diamA) n 2nr , (9) where αn is the volume of the n-dimensional unit ball [1].”