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 $c{r}^{d-3}$, where $c$ is an intrinsic power volume of $C$.