The derivative of the parallel volume difference

Abstract

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 r → ∞ at order r −2 . We will use this result to examine Brownian paths and Boolean models.

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

“…Proof. Kampf proved in [15], lemma 28, that for every compact set A there exists a constant C which depends on n, A so that for every t ≥ 1,…”

“…2 by a convex body B = rB n 2 + M , for some r > 0 and some convex body M such that its support function h B (u) = max{< x, u >, x ∈ B} is twice differentiable on R n \{0} because inequality (6) of [15] holds with these assumptions.…”

Large parallel volumes of finite and compact sets in d-dimensional Euclidean space

“…We discuss mainly the parallel volume of finite sets, which turns out to be already nontrivial, but Theorem 2, below, deals with compact sets C ⊆ R d . A first result for large parallel volumes has been obtained in [4] for compact sets C ⊆ R d , where it is shown that the volume of C r is close to the volume of its convex hull conv C r = (conv C) r . In fact, there is a constant…”

“…for all sufficiently large r. In [4] an example set C was given where this volume difference behaves like r d−3 , so the exponent here is best possible. Independently, a weaker version of (3) was shown in [7, Lemma 2], where C was assumed to be an at most two-dimensional subset of the set {0, 1} d , d ≥ 3, and the volume difference was shown to converge to zero faster than r d−2 .…”

“…Other applications of our expansions appear to be in reach. For example, in [4] formula (3) was used to examine the expected value of the parallel volume of Brownian paths, when the time is small, and to relate analytical properties of r → W (rK) to geometric properties of K, where W denotes the Wills functional and K is a fixed compact set.…”