Contact distributions of Boolean models (Q2704106)

Contact distributions are an important tool for the description and analysis of spatial random structures in \(\mathbb{R}^d\). The contact distribution of a random closed set \(Z\) is the conditional distribution of the distance from a reference point \(x\in\mathbb{R}^d\) to \(Z\) given that \(x\not\in Z\). Distances are measured with respect to a suitable gauge body \(B\) which, in the present paper, is assumed to be a compact convex set (convex body) with \(0\in B\).NEWLINENEWLINENEWLINEInstead of considering the distribution of the distance from a reference point \(x\), more generally the author studies the distribution of the nearest point in \(Z\) measured from \(x\) (the local contact distribution), where \(Z\) is a (stationary) Boolean model with polyconvex grains (finite unions of convex bodies). Such a Boolean model can be obtained as the union set of a Poisson particle process \(X\). A major aim in the present contribution is to establish relations between characteristics of \(X\) such as the intensity, the spatial density in the instationary case or the shape distribution and the local contact distribution functions of \(Z\). For instance, it is shown how the mean surface area measure of the typical grain of \(X\) can be determined by means of dilatation volumes of \(Z\). The results and proofs in the paper essentially use certain nonnegative extensions of support measures.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00040].