Abstract

Let XX be a separable superreflexive Banach space and ff be a semiconvex function (with a general modulus) on XX. For k∈Nk \in \N, let Σk(f)\Sigma_k(f) be the set of points x∈Xx\in X, at which the Clarke subdifferential ∂f(x)\partial f(x) is at least kk-dimensional. Note that Σ1(f)\Sigma_1(f) is the set of all points at which ff is not Gâteaux differentiable. Then Σk(f)\Sigma_k(f) can be covered by countably many Lipschitz surfaces of codimension kk which are described by functions, which are differences of two semiconvex functions. If XX is separable and superreflexive Banach space which admits an equivalent norm with modulus of smoothness of power type 22 (e.g., if XX is a Hilbert space or X=Lp(μ)X=L^p(\mu) with 2≤p2 \leq p), we give, for a fixed modulus ω\omega and k∈Nk \in \N, a complete characterization of those A⊂XA\subset X, for which there exists a function ff on XX which is semiconvex on XX with modulus ω\omega and A⊂Σk(f)A \subset \Sigma_k(f). Namely, A⊂XA\subset X has this property if and only if AA can be covered by countably many Lipschitz surfaces SnS_n of codimension kk which are described by functions, which are differences of two Lipschitz semiconvex functions with modulus CnωC_n \omega.

Author information

Contact details are reproduced from the original publication and may be historical.

Luděk Zajíček

Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

zajicek@karlin.mff.cuni.cz

Keywords

  • Semiconvex function with general modulus
  • Clarke subdifferential
  • singular set
  • singular point of order k
  • Lipschitz surface
  • DSC surface
  • superreflexive space

Mathematics Subject Classification

49J52; 46G05

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Suggested citation

J. Duda, L. Zajíček. “Smallness of Singular Sets of Semiconvex Functions in Separable Banach Spaces.” Journal of Convex Analysis 20 (2013), No. 2, 573–598. https://doi.org/10.68381/jca20034

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press LLC.