On the singularities of convex functions
A rectifiability result is provided for the singular sets of convex and semiconvex functions. In fact, for every real convex or semiconvex function \(u\) on a convex open subset \(\Omega\) of \({\mathbf R}^ n\), and every integer \(k\) such that \(0< k\leq n\), one may consider the set \(\Sigma^ k\) of all points \(x\in\Omega\) such that the subdifferential of \(u\) in \(x\) has dimension greater than or equal to \(k\). Then the set \(\Sigma^ k\) is countably \((n-k)\)-rectifiable, and this means that it may be covered by countably many \((n-k)\)-dimensional submanifolds of class \(C^ 1\), except for a \({\mathcal H}^{n-k}\) negligible subset, where \({\mathcal H}^{n-k}\) denotes the \((n-k)\)-dimensional Hausdorff measure.
singular sets of convex and semiconvex functions
scientific article; zbMATH DE number 219912
scientific article