Absolute Continuity for Curvature Measures of Convex Sets, III

Abstract

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 À 1Þ-dimensional Hausdorff measure in Euclidean space R d : 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 À 1 À rÞth surface area measure of the polar body, and conversely. Applications are also considered. # 2002 Elsevier Science (USA)

Random points on the boundary of smooth convex bodies

“…Using a local version of an integral geometric projection formula for surface area measures which is contained in Hug [11], Lemma 4.3, this equals…”

Hessian Measures of Convex Functions and Applications to Area Measures

“…A special case of (3.12) yields that Theorem 5 can be used to give a simplified proof (on the basis of the results for Hessian measures) of a theorem due to Weil [19], which concerns surface area measures of convex bodies. A corresponding result for curvature measures has been deduced recently in [12] from the one for surface area measures. However, it does not seem to be possible to derive the result for curvature measures directly from Theorem 5.…”

“…. , d} and H d -almost everywhere, the Radon-Nikodym [12]. Roughly speaking, for curvature measures the result states that the absolute continuity of the mean curvature measure of a convex set and some integrability assumption for the mean curvature (in Aleksandrov's sense) together imply the absolute continuity of further (lower order) curvature measures of the given set.…”

“…In the case of Hessian measures, the result can be described as follows. Assume that the Hessian measure F d−1 (u, ·) of a (semi-)convex function u, defined in Ω, is absolutely continuous with respect to H d and that, for some p 2, [12] that this result (in a sense) is best possible. As a consequence, we obtain Weil's theorem for area measures by applying our result in the special case of Hessian measures of support functions.…”

Some aspects of the variational nature of mean curvature flow

“…We expect that Theorem 4.1 holds with the same proof in n = 3 space dimensions, on using the results of[21]-[23].…”