Maps with the Radon-Nikod\'ym property

TL;DR

This paper explores the extension of the Radon-Nikodým property to metric spaces via dentable maps, establishing foundational results and examining implications for approximation and variational principles.

Contribution

It introduces a generalized framework for the Radon-Nikodým property using dentable maps and analyzes their properties and applications in Banach spaces.

Findings

The dual elements strongly slicing form a dense G_delta subset.

The space of uniformly continuous dentable maps is a Banach space.

Classical variational principles do not extend beyond traditional hypotheses.

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-Nikod\'ym 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 $\mathcal{G}_\delta$ 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.