Hessian measures of semi-convex functions and applications to support measures of convex bodies

This paper originates from the investigation of support measures of convex bodies (sets of positive reach), which form a central subject in convex geo metry and also represent an important tool in related fields. We show that these measures are absolutely continuous with respect to Hausdorff measures of appropriate dimensions, and we determine the Radon-Nikodym derivatives explicitly on sets of sigma-finite Hausdorff measure. The results which we obtain in the setting of the theory of convex bodies (sets of positive reac h) are achieved as applications of various new results on Hessian measures of convex (semi-convex) functions. Among these are a Crofton formula, resul ts on the absolute continuity of Hessian measures, and a duality theorem wh ich relates the Hessian measures of a convex function to those of the conju gate function. In particular, it turns out that curvature and surface area measures of a convex body K are the Hessian measures of special functions, namely the distance function and the support function of K.