The parallel volume at large distances

Abstract

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.

On the analogue of the concavity of entropy power in the Brunn–Minkowski theory

“…Notice that this complements the result of Kampf [14] who proved that V conv(A) (t) − V A (t) tends to 0 as t → +∞.…”

“…is convex as the sum of an affine function and a convex function. Notice that this complements the result of Kampf [14] who proved that V conv(A) (t) − V A (t) tends to 0 as t → +∞.…”

On the improvement of concavity of convex measures

“…For more modern applications, the parallel volume and its generalized forms are still studied (see e.g. [12], [13]). Moreover, this notion of parallel volume leads to the powerful theory of mixed volumes (see [22] for further details).…”

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 and . 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 .…”