On the structure of singular sets of convex functions (Q1315028)

Let \(f: \mathbb{R}^ h\to \mathbb{R}\) be a convex function, let \(k\) be an integer such that \(0< k< h\), and let \(\Sigma^ k(f)\) be defined by \[ \Sigma^ k(f)= \bigl\{x\in \mathbb{R}^ h: \dim \partial f(x)\geq k\bigr\}, \] where \(\partial f\) is the subdifferential of \(f\). It is proved that there exists a sequence \((M_ i)_{i\in\mathbb{N}}\) of \((h-k)\)-dimensional submanifolds of class \(C^ 2\) in \(\mathbb{R}^ h\) such that \[ {\mathcal H}_{h- k}\left(\Sigma^ k(f)\backslash\bigcup_{i\in\mathbb{N}} M_ i\right)=0, \] where \({\mathcal H}_{h-k}\) denotes the usual \((h-k)\)-dimensional Hausdorff measure in \(\mathbb{R}^ h\).