Mean Densities and Spherical Contact Distribution Function of Inhomogeneous Boolean Models

Abstract

The mean density of a random closed set Θ in ℝ d with Hausdorff dimension n is the Radon–Nikodym derivative of the expected measure 𝔼[ℋ n (Θ ∩·)] induced by Θ with respect to the usual d-dimensional Lebesgue measure. Starting from an open problem posed by Matheron in [24, pp. 50–51], we consider here inhomogeneous Boolean models Ξ in ℝ d with integer Hausdorff dimension n ∈ {0,…, d}, and we study the mean density of their boundary (which is their mean density if n < d) and the differentiability of their spherical contact distribution function H Ξ, under general regularity assumptions on the typical grain, related to the existence of its (outer) Minkowski content. In particular, we provide an explicit formula for ∂2 H Ξ(r, x)/(∂r 2) at r = 0 for a class of Boolean models, whose typical grain has positive reach; known results for stationary Boolean models with convex grains follows then as a particular case. Examples and statistical applications are also discussed.