On the singularities of convex functions
type: Item
instance of: scholarly article
title: On the singularities of convex functions
zbMATH Open document ID: 0784.49011
DOI: 10.1007/BF02567770
published in: Manuscripta Mathematica
publication date: 1993-06-29T00:00:00Z
full work available at URL: https://eudml.org/doc/155763
review text: 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.
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 26B25
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 219912
zbMATH Keywords: Hamilton-Jacobi equations
zbMATH Keywords: nonsmooth analysis
zbMATH Keywords: singular sets of convex and semiconvex functions
zbMATH Keywords: subdifferential
zbMATH Keywords: Hausdorff measure
author: Giovanni Alberti
author: Luigi Ambrosio
author: Piermarco Cannarsa
MaRDI profile type: Publication
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OpenAlex ID: W1991408703
description: scientific article; zbMATH DE number 219912