Given a (semi)-convex function $u : \Omega\subset R^n \to R$ and an integer $k$ in $[0, n]$, we show that the set $\Sigma^k$ of all points $x$ where the dimension of the subdifferential of $f$ is larger or equal than $k$ is countably ${\cal H}^{n-k}$-rectifiable, i.e., it is contained (up to a ${\cal H}^{n-k}$-negligible set) in a countable union of $C^1$ hypersurfaces of dimension $n-k$.
On the singularities of convex functions
ALBERTI, GIOVANNI;
1992-01-01
Abstract
Given a (semi)-convex function $u : \Omega\subset R^n \to R$ and an integer $k$ in $[0, n]$, we show that the set $\Sigma^k$ of all points $x$ where the dimension of the subdifferential of $f$ is larger or equal than $k$ is countably ${\cal H}^{n-k}$-rectifiable, i.e., it is contained (up to a ${\cal H}^{n-k}$-negligible set) in a countable union of $C^1$ hypersurfaces of dimension $n-k$.File in questo prodotto:
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