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Zbl 0784.49011
Alberti, G.; Ambrosio, L.; Cannarsa, P.
On the singularities of convex functions. (English)
[J] Manuscr. Math. 76, No.3-4, 421-435 (1992). ISSN 0025-2611; ISSN 1432-1785/e

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 ${\bold R}\sp n$, and every integer $k$ such that $0< k\le n$, one may consider the set $\Sigma\sp k$ 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 $\Sigma\sp k$ is countably $(n-k)$-rectifiable, and this means that it may be covered by countably many $(n-k)$-dimensional submanifolds of class $C\sp 1$, except for a ${\cal H}\sp{n-k}$ negligible subset, where ${\cal H}\sp{n-k}$ denotes the $(n-k)$-dimensional Hausdorff measure.
[G.Alberti (Pisa)]

MSC 2000:: *49J52 Nonsmooth analysis (other weak concepts of optimality)
26B25 Convexity and generalizations (several real variables)
28A78 Hausdorff measures

Keywords: Hamilton-Jacobi equations; nonsmooth analysis; singular sets of convex and semiconvex functions; subdifferential; Hausdorff measure

Cited in: Zbl 0923.49010 Zbl 0874.49041

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