2009/05/31 by Jan Rataj, Steffen Winter · 47 citations
Mathematics · #Brownian motion #Combinatorics #Euclidean geometry #Euclidean space #Fractal #Geometry #Limit (mathematics) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Minkowski addition #Minkowski space #Point processes and geometric inequalities #Pure mathematics #Set (abstract data type) #Space (punctuation) #Statistics #Stochastic processes and statistical mechanics #Surface (topology) #Trajectory #Volume (thermodynamics) #math.CA #msc:28A75 #msc:52A20 #msc:60D05
paper · pdf · doi:10.1512/iumj.2010.59.4165
published in Indiana University Mathematics Journal 59(5), 1661-1686 (Indiana University) · 19 pages
openalex publication_date 2010/01/01 · arxiv created 2010/10/11 · arxiv updated 2011/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.