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

For a given convex (semi-convex) function $u$, defined on a nonempty open convex set $\Omega\subset\mathbf{R}^n$, 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). We prove that, for $r\in \{0,\ldots,n\}$, the $r$-th coefficient measure of the local Steiner formula for $u$, restricted to the set of $r$-singular points of $u$, is absolutely continuous with respect to the $r$-dimensional Hausdorff measure, and that its density is the $(n-r)$-dimensional Hausdorff measure of the subgradient of $u$.

As an application, under the assumptions that $u$ is convex and Lipschitz, and $\Omega$ is bounded, we get sharp estimates for certain weighted Hausdorff measures of the sets of $r$-singular points of $u$. Such estimates depend on the Lipschitz constant of $u$ and on the quermassintegrals of the topological closure of $\Omega$.