Maps with the Radon–Nikodým Property

Abstract

Abstract. We study dentable maps from a closed convex subset of a Banach space into a metric space as an attempt of generalize the Radon-Nikodým property to a "less linear" frame. We note that a certain part of the theory can be developed in rather great generality. Indeed, we establish that the elements of the dual which are "strongly slicing" for a given uniformly continuous dentable function form a dense G δ subset of the dual. As a consequence, the space of uniformly continuous dentable maps from a closed convex bounded set to a Banach space is a Banach space. However some interesting applications, as Stegall's variational principle, are no longer true beyond the usual hypotheses, sending us back to the classical case. Moreover, we study the connection between dentability and approximation by delta-convex functions for uniformly continuous functions. Finally, we show that the dentability of a set is closely related with the dentability of delta-convex maps defined on it.

Some stability properties for the Bishop–Phelps–Bollobás property for Lipschitz maps

“…In addition, it is shown that if (Γ, w * ) is discrete then every bounded operator T ∈ L(X, Y ) is Γ -flat. Let us also mention that the recently introduced notion of dentable map [11] implies Γ -flatness.…”

On uniformly convex functions

“…Finally we will discuss the approximation by differences of convex functions in terms of the index of dentability improving [25,Theorem 1.4] and [15,Theorem 4.1]. A real function defined on a convex set is called DC-Lipschitz if it is the difference of two convex Lipschitz functions.…”