The parallel volume at large distances (Q715179)

In the \(d\)-dimensional Euclidean space \(\mathbb{R}^d\), a body is defined to be a non-empty compact set. The distance from a point \(x\) to a body \(K\) is the infimum of all Euclidean distances from \(x\) to points of \(K\). The Lebesgue measure of the parallel body is called the parallel volume of \(K\) at distance \(r\). In this paper, the author considers the asymptotic behaviour of the parallel volume of planar non-convex bodies as the distance tends to infinity. For a body \(K \subseteq \mathbb{R}^2\) it is shown that \[ \lim_{r\to\infty}V2((\mathrm{conv} K) + r B^2)-V2(K + r B^2) = 0. \] In fact the formula is the two dimensional deterministic special case of a more general result holding for random bodies in \(\mathbb{R}^d\), \(d\geq 2\), which also yields a new proof for the fact that the parallel volume of a body \(K\subseteq \mathbb{R}^2\) is a polynomial if and only if \(K\) is convex. Motivations and interesting extensions to Minkowski spaces and random sets are also discussed.