On the Second Derivative of the Spherical Contact Distribution Function of Smooth Grain Models

We consider a stationary grain model \Xi with convex, compact and smoothly bounded grains. We prove that the density f of the spherical contact distribution of \Xi is absolutely continuous and derive an explicit formula for the corresponding Radon-Nikodym derivative f 0 . The value f 0 (0) is of a simple form and admits a nice geometric interpretation. SPHERICAL CONTACT DISTRIBUTION; GERM-GRAIN MODEL; PALM PROBABILITY; SURFACE MEASURE; RANDOM SET; RANDOM MEASURE; CCURVATURE MEASURES AMS 1991 SUBJECT CLASSIFICATIONS: PRIMARY 60G57, SECONDARY 60D05 1 Introduction We consider a stationary random closed set \Xi = 1 [ n=1 \Xi n ; defined on the probability space(\Omega ; F ; P ), where the \Xi n , n 2 N , are random non-empty compact and convex sets in R d , d 2, with a smooth boundary. We assume that the volume fraction p := P (0 2 \Xi) is less than 1 and denote by d(x) the distance from x 2 R d to \Xi. The conditional distribution function F of d(x), given that x = 2 \Xi,..