Mixed Measures of Convex Cylinders and Quermass Densities of Boolean Models

Abstract

Translative integral formulas for curvature measures of convex bodies were obtained by Schneider and Weil by introducing mixed measures of convex bodies. These results can be extended to arbitrary closed convex sets since mixed measures are locally defined. Furthermore, iterated versions of these formulas due to Weil were used by Fallert to introduce quermass densities for (non-stationary and non-isotropic) Poisson processes of convex bodies and respective Boolean models. In the present paper, we first compute the special form of mixed measures of convex cylinders and prove a translative integral formula for them. After adapting some results for mixed measures of convex bodies to this setting we then use this integral formula to obtain quermass densities for (non-stationary and non-isotropic) Poisson processes of convex cylinders. Furthermore, quermass densities of Boolean models of convex cylinders are expressed in terms of mixed densities of the underlying Poisson process generalizing classical formulas by Davy and recent results by Spiess and Spodarev.