Curvature measures and random sets II

Abstract

In choosing models of stochastic geometry three general problems play a role which are closely connected with each other: In the present paper second order local geometric properties of random subsets of R d are of interest. These properties are described by signed curvature measures in a measure geometric context. The theory of point processes on general spaces (here on the space of subsets with positive reach) provides an appropriate framework for solving construction and measurability problems. Mean value relations for random curvature measures associated with such set processes are derived by means of invariance properties. Ergodic interpretations of the curvature densities are also given. The appendix provides auxiliary results for random signed Radon measures in locally compact separable Hausdorff spaces.