The parallel volume at large distances

In this paper we examine the asymptotic behavior of the parallel volume of planar non-convex bodies as the distance tends to infinity. We show that the difference between the parallel volume of the convex hull of a body and the parallel volume of the body itself tends to 0. This yields a new proof for the fact that a planar body can only have polynomial parallel volume if it is convex. Extensions to Minkowski spaces and random sets are also discussed.