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Maps with the Radon-Nikodým property. (English) Zbl 1396.46013
Let $$X$$ be a real Banach space with dual $$X^*$$, $$C$$ a closed convex subset of $$X$$ and $$(M,d)$$ a metric space. A function $$f:C\to M$$ is called dentable if, for every nonempty bounded subset $$A$$ of $$C$$ and every $$\varepsilon>0$$, there exists an open half-space $$H$$ of $$X$$ such that $$A\cap H\neq\emptyset$$ and diam$$(f(A\cap H))<\varepsilon$$. Denote by $$\mathcal D(C,M)$$ the set of all dentable mappings from $$C$$ to $$M$$ and by $$\mathcal D_U(C,M)$$ its subset formed by all dentable mappings uniformly continuous on bounded subsets of $$C$$.
The notion is related to the Radon-Nikodým (RN) property: the set $$C$$ has the RN property iff the identity mapping $$I:C\to(C,\|\cdot\|)$$ is dentable. Also, a continuous linear operator from $$X$$ to another Banach space $$Y$$ is dentable iff it is an RN operator in the sense of O. I. Reinov [Sov. Math., Dokl. 16, 119–123 (1975; Zbl 0317.47022); translation from Dokl. Akad. Nauk SSSR 220, 528–531 (1975)], and so the study of dentable mappings is, in some sense, a nonlinear extension of the RN property. The authors show that the set $$C$$ has the RN property iff every Lipschitz mapping $$f:C\to M$$ is dentable. If $$M$$ is a Banach space (Banach algebra, Banach lattice), then $$\mathcal D_U(C,M)$$ is also a Banach space (Banach algebra, Banach lattice, respectively) with respect to the norm of uniform convergence on $$C$$.
It is known that the strongly exposing functionals on a closed convex set with the RN property form a dense $$G_\delta$$ subset of $$X^*$$. The authors extend this result to this frame by replacing strongly exposing functionals by a class of functionals called strongly slicing. The possibility of uniform approximation of a uniformly continuous function $$f$$ by DC (difference of convex) functions is also studied. It turns out that this happens iff the function $$f$$ is finitely dentable in the sense defined by M. Raja [J. Convex Anal. 15, No. 2, 219–233 (2008; Zbl 1183.46018)]. Other results as, for instance, Stegall’s variational principle, are no longer true beyond the usual hypotheses, sending back to the classical case.
##### MSC:
 46B22 Radon-Nikodým, Kreĭn-Milman and related properties 46E40 Spaces of vector- and operator-valued functions 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 46T20 Continuous and differentiable maps in nonlinear functional analysis