On the singularities of convex functions

Given a semi-convex functionu: ω⊂R n→R and an integerk≡[0,1,n], we show that the set ∑k defined by $$\Sigma ^k = \left\{ {x \in \Omega :dim\left( {\partial u\left( x \right)} \right) \geqslant k} \right\}$$ is countably ℋn-k i.e., it is contained (up to a ℋn-k-negligible set) in a countable union ofC 1 hypersurfaces of dimensions (n−k). Moreover, we show that $$\int\limits_{\Omega ' \cap \Sigma ^k } {\mathcal{H}^k } \left( {\partial u\left( x \right)} \right)d\mathcal{H}^{n - k} \left( x \right)< + \infty $$ for any open set ω′⊂⊂ω.