Large parallel volumes of finite and compact sets in d-dimensional Euclidean space

Abstract

The r-parallel volume V (C r ) of a compact subset C in d-dimensional Euclidean space is the volume of the set C r of all points of Euclidean distance at most r > 0 from C. According to Steiner's formula, V (C r ) is a polynomial in r when C is convex. For finite sets C satisfying a certain geometric condition, a Laurent expansion of V (C r ) for large r is obtained. The dependence of the coefficients on the geometry of C is explicitly given by so-called intrinsic power volumes of C. In the planar case such an expansion holds for all finite sets C. Finally, when C is a compact set in arbitrary dimension, it is shown that the difference of large r-parallel volumes of C and of its convex hull behaves like cr d−3 , where c is an intrinsic power volume of C.