On representation of boundary integrals involving the mean curvature for mean-convex domains

Given a mean-convex domain $\Omega\subset \R^n$ with boundary of class $C^{2,1}$, we provide a representation formula for a boundary integral of the type \[ \int_{\partial \Omega} f(k(x)) \, d\mathcal{H}^{n-1} \] where $k\geq 0$ is the mean curvature of $\partial \Omega$ and $f$ is non-increasing and sufficiently regular, in terms of volume integrals and defect measure on the ridge set.