
Journal of Convex Analysis 20 (2013), No. 2, 573598 Copyright Heldermann Verlag 2013 Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces Jakub Duda 346 West 56th Street, New York, NY 10019, U.S.A. jakub.duda@gmail.com Ludek Zajícek Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic zajicek@karlin.mff.cuni.cz [Abstractpdf] \newcommand{\N}{{\mathbb N}} Let $X$ be a separable superreflexive Banach space and $f$ be a semiconvex function (with a general modulus) on $X$. For $k \in \N$, let $\Sigma_k(f)$ be the set of points $x\in X$, at which the Clarke subdifferential $\partial f(x)$ is at least $k$dimensional. Note that $\Sigma_1(f)$ is the set of all points at which $f$ is not G\^ ateaux differentiable. Then $\Sigma_k(f)$ can be covered by countably many Lipschitz surfaces of codimension $k$ which are described by functions, which are differences of two semiconvex functions. If $X$ is separable and superreflexive Banach space which admits an equivalent norm with modulus of smoothness of power type $2$ (e.g., if $X$ is a Hilbert space or $X=L^p(\mu)$ with $2 \leq p$), we give, for a fixed modulus $\omega$ and $k \in \N$, a complete characterization of those $A\subset X$, for which there exists a function $f$ on $X$ which is semiconvex on $X$ with modulus $\omega$ and $A \subset \Sigma_k(f)$. Namely, $A\subset X$ has this property if and only if $A$ can be covered by countably many Lipschitz surfaces $S_n$ of codimension $k$ which are described by functions, which are differences of two Lipschitz semiconvex functions with modulus $C_n \omega$. Keywords: Semiconvex function with general modulus, Clarke subdifferential, singular set, singular point of order k, Lipschitz surface, DSC surface, superreflexive space. MSC: 49J52; 46G05 [ Fulltextpdf (242 KB)] for subscribers only. 