On the second derivative of the spherical contact distribution function ofsmooth grain models

We consider a stationary grain model Xi in R-d with convex, compact and smo othly bounded grains. We study the spherical contact distribution function F of Xi and derive (under suitable assumptions) an explicit formula for its second derivative F ". The value F " (0) is of a simple form and admits a clear geometric interpretation. For the Boolean model we obtain an interest ing new formula for the (d - 1)-st quermass density.