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

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} r d − 2

in a

d 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} r d − 3

. We will characterize the

2 2

-dimensional Minkowski spaces in which the parallel volume difference has always at most order

r^{-1} r − 1

. Finally we present applications concerning Brownian paths and Boolean models.