Steiner type formulae and weighted measures of singularities of semi-convex functions

For a given convex (semi-convex) function u, dened on a nonempty open convex set of the n dimensional Euclidean space, we establish a local Steiner type formula, the coefficients of which are nonnegative (signed) Borel measures. We also determine explicit integral representations for these coefficient measures, which are similar to the integral representations for the curvature measures of convex bodies (and, more generally, of sets with positive reach).