This is an announcement for the paper “Maps with the Radon-Nikodým property” by Luis García-Lirolahttps://arxiv.org/find/math/1/au:+Garcia_Lirola_L/0/1/0/all/0/1, Matías Rajahttps://arxiv.org/find/math/1/au:+Raja_M/0/1/0/all/0/1.
Abstract: We study dentable maps from a closed convex subset of a Banach space into a metric space as an attempt of generalise 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 $G_{\delta}$ 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 relation between dentable maps and delta-convex maps.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.08585