Absolute continuity for curvature measures of convex sets

This 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 ðd 1Þ-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 ðd 1 rÞth surface area measure of the polar body, and conversely. Applications are also considered. # 2002 Elsevier Science (USA) Key Words: curvature measures; surface area measures; absolute continuity of measures; bounded density; polar body; curvatures on the unit normal bundle; integral geometry; stability results. 1