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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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