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Finitely dentable functions, operators and sets. (English) Zbl 1183.46018
A map $$f$$ from a closed convex bounded subset $$C$$ of a Banach space $$X$$ to a metric space $$Y$$ is said to be finitely dentable if for any $$\varepsilon>0$$, the Cantor-Bendixson derivation which consists in removing all slices on which the oscillation of $$f$$ is less than $$\varepsilon$$ terminates on the empty set after finitely many steps. It is shown that a bounded linear operator is uniformly convexifying in Beauzamy’s sense if and only if its restriction to the unit ball is finitely dentable.
Another very interesting result, which extends a theorem due to M. Cepedello, states that a real-valued Lipschitz function defined on $$C$$ is a uniform limit of differences of convex Lipschitz functions if and only if it is finitely dentable.
Finally, the set $$C$$ itself is said to be finitely dentable if the identity map is finitely dentable. It is shown that such a set $$C$$ is weakly compact, uniformly Eberlein in its weak topology, and that there is a reflexive space $$E$$ and a uniformly convexifying bounded operator from $$E$$ to $$X$$ such that $$C$$ is contained in $$T(B_E)$$. The proofs use in particular G. Lancien’s proof of the Enflo-Pisier renorming theorem, and infimum-convolution arguments.

MSC:
 46B22 Radon-Nikodým, Kreĭn-Milman and related properties 46B03 Isomorphic theory (including renorming) of Banach spaces 46T20 Continuous and differentiable maps in nonlinear functional analysis
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