Generalized curvature measures and singularities of sets with positive reach

For a convex body K in Euclidean space R-d (d greater than or equal to 2) a nd for r is an element of {0,...,d - 1}, let Sigma(r)(k') be the set of r-s ingular boundary points of K. It is known that Sigma(r)(K) is countably r-r ectifiable and hence has sigma-finite r-dimensional Hausdorff measure. We o btain a quantitative improvement of this result, taking into account the st rength of the singularities. Denoting by Sigma(r)(K, tau) the set of those r-singular boundary points of K at which the spherical image has (d - 1 - r )-dimensional Hausdorff measure at least tau > 0, we establish a finite upp er bound for the r-dimensional Hausdorff measure of Sigma(r)(K, tau). This estimate is deduced from an identity that connects Hausdorff measures of sp herical images of singularities to the generalized curvature measure of the convex body K. The latter relation is, in fact, proved for the class of se ts with positive reach. For convex bodies, similar result as for singular b oundary points are obtained for singular normal vectors. We also consider t he disintegration of generalized curvature measures with respect to project ions onto the components of the product space R-d x Sd-1.