The decompositions of curvature measures via Hausdorff measures on singular sets of convex bodies

Abstract

This paper proves decomposition formulas for the curvature measures of convex bodies. The decomposition of a curvature measure of a convex body is with respect to Hausdorff measures of different dimensions restricted to the singular sets of the boundary of the convex body. The density functions and singular measures in the decomposition are explicitly given in terms of integrals of functions of the generalized curvatures of the convex body. A similar decomposition for curvature measures defined on the unit sphere is also proved.