Absolute Continuity for Curvature Measures of Convex Sets, III

AbstractThis work is devoted to the investigation of the basic relationship between the geometric shape of a convex set and measure theoretic properties of the associated curvature and surface area measures. We study geometric consequences of and conditions for absolute continuity of curvature and surface area measures with respect to (d−1)-dimensional Hausdorff measure in Euclidean space Rd. Our main results are two “transfer principles” which allow one to translate properties connected with the absolute continuity of the rth curvature measure of a convex body to dual properties related to the absolute continuity of the (d−1−r)th surface area measure of the polar body, and conversely. Applications are also considered