Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces

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 $C_n \omega$.