On the singularities of convex functions

Abstract

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 ℋ^n-k i.e., it is contained (up to a ℋ^n-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.