Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces

Let

be a separable superreflexive Banach space and

f f

be a semiconvex function (with a general modulus) on

X X

. For

k \in {{\mathbb N}} k ∈ N

, let

\Sigma_k(f) Σ k ( f )

be the set of points

x\in X x ∈ X

, at which the Clarke subdifferential

\partial f(x) ∂ f ( x )

is at least

k k

-dimensional. Note that

\Sigma_1(f) Σ 1 ( f )

is the set of all points at which

f f

is not Gâteaux differentiable. Then

\Sigma_k(f) Σ k ( f )

can be covered by countably many Lipschitz surfaces of codimension

k k

which are described by functions, which are differences of two semiconvex functions. If

X X

is separable and superreflexive Banach space which admits an equivalent norm with modulus of smoothness of power type

2 2

(e.g., if

X X

is a Hilbert space or

X=L^p(\mu) X = L p ( μ )

with

2 \leq p 2 ≤ p

), we give, for a fixed modulus

\omega ω

and

k \in {{\mathbb N}} k ∈ N

, a complete characterization of those

A\subset X A ⊂ X

, for which there exists a function

f f

on

X X

which is semiconvex on

X X

with modulus

\omega ω

and

A \subset \Sigma_k(f) A ⊂ Σ k ( f )

. Namely,

A\subset X A ⊂ X

has this property if and only if

A A

can be covered by countably many Lipschitz surfaces

S_n S n

of codimension

k k

which are described by functions, which are differences of two Lipschitz semiconvex functions with modulus

C_n \omega C n ω

.