Polynomial parallel volume, convexity and contact distributions of random sets

Abstract

We characterize convexity of a random compact set X in R d via polynomial expected parallel volume of X. The parallel volume of a compact set A is a function of r ≥ 0 and is defined here in two steps. First we form the parallel set at distance r with respect to a oneor two-dimensional gauge body B. Then we integrate the volume of this (relative) parallel set with respect to all rotations of B. We apply our results to characterize convexity of the typical grain of a Boolean model via first contact distributions.

The parallel volume at large distances

“…This result was generalized by Hug, Last and Weil [3] to more general 2-dimensional Minkowski spaces and to random sets. Moreover they gave a interpretation of the result for higher-dimensional bodies.…”

“…Hug, Last and Weil [3] extended this result to random bodies and more general 2-dimensional normed spaces. They also gave an interpretation for bodies K having more than 2 dimensions.…”

“…We will now give a new proof of this theorem under assumptions similar those of made in [3]. Then we will discuss applications to the Wills functional and its generalisation introduced in [4], namely the functionals f µ .…”

“…In section 4 we give a new proof for Theorem 1.2 under generalisations quite similar to those of [3].…”

On the Specific Area of Inhomogeneous Boolean Models. Existence Results and Applications

“…This is true for instance when Θ satisfies a (local) Steiner formula; in this case the limit in Eq. 5 can be studied in terms of the quermass densities (or Minkowski functionals) associated to Θ, and so by means of tools from integral geometry mainly Weil, 2001;Hug et al, 2006;Baddeley et al, 2007, and references therein). For other related works see also Hug and Last (2000), Hug et al (2004), and Kiderlen and Rataj (2006).…”