The derivative of the parallel volume difference

In this paper we continue the investigation of the asymptotic behavior of the parallel volume in Minkowski spaces as the distance tends to infinity that was started in 8 and 9 . Our main result is that the derivative of the difference between the parallel volume of the convex hull of a planar body and the parallel volume of the body itself tends to 0 for at order . We will use this result to examine Brownian paths and Boolean models.