Intersection Formulae of Integral Geometry 21

We establish extensions of the Crofton formula and, under some restrictions, of the principal kinematic formula of integral geometry from curvature measures to generalized curvature measures of convex bodies. We also treat versions for nite unions of convex bodies. As a consequence, we get a new intuitive interpretation of the area measures of Aleksandrov and Fenchel{ Jessen. The subject of this paper is the generalization of two integral geometric intersection formulae for curvature measures of convex bodies: the Crofton formula and the principal kinematic formula. The curvature measures of Federer are replaced by the so-called generalized curvature measures, which are concentrated on the set of all support elements of a convex body, and which for this reason we will also call support measures. The proofs of these extensions depend on certain easy-to-state assertions about the boundary structure of convex bodies. For one of these assertions , we were only able to give a proof under some restrictions on the convex bodies under consideration. This results in corresponding limitations for our generalized principal kinematic formula. We strongly conjecture that in fact these restrictions are not necessary. Our version of the Crofton formula, which can be proved without any restrictions, gives rise to a new intuitive interpretation of the support measures and especially of the area measures of convex bodies. We also treat extensions to the convex ring, the set of all nite unions of convex bodies. For analogous results in spherical space, see my thesis 1], which also contains the results of the present article. The paper 2] is a summary of 1]. It remains open whether there exist extensions to classes of more general sets, as considered in the case of the curvature measures, e.g., by Rother & ZZ ahle 3]. For a recent survey on integral geometry of convex bodies, see Schneider & Wieacker 7].