Asymptotic Order of the Parallel Volume Difference in Minkowski Spaces

We investigate the asymptotic behavior of the parallel volume of fixed non-convex bodies in Minkowski spaces as the distance $r$ tends to infinity. We will show that the difference of the parallel volume of the convex hull of a body and the parallel volume of the body itself, which is called parallel volume difference, can at most have order $r^{d-2}$ in a $d$-dimensional Minkowski space. Then we will show that in certain Minkowski spaces (and in particular in Euclidean spaces) this difference can at most have order $r^{d-3}$. We will characterize the $2$-dimensional Minkowski spaces in which the parallel volume difference has always at most order $r^{-1}$. Finally we present applications concerning Brownian paths and Boolean models.