On the structure of non dentable subsets of C({\omega}^{\omega}^k)

TL;DR

This paper investigates the structure of non-dentable subsets in certain function spaces, showing limitations on their properties and establishing equivalences between the Radon-Nikodym property and the KMP on these subsets.

Contribution

It proves the non-existence of certain non-dentable convex subsets with specific property equivalences and links the RNP and KMP on subsets of C({low}^k).

Findings

No K closed convex bounded non-dentable subset with PCP and RNP equivalence exists.

Every non-dentable subset contains a non-dentable subset where weak and norm topologies coincide.

RNP and KMP are equivalent on subsets of C({low}^k).

Abstract

It is shown that there is no K closed convex bounded non-dentable subset of C({\omega}^{\omega} ^k) such that on the subsets of K the PCP and the RNP are equivalent properties. Then applying Schachermayer-Rosenthal theorem, we conclude that every non-dentable K contains non-dentable subset L so that on L the weak topology coincides with the norm one. It follows from known results that the RNP and the KMP are equivalent properties on the subsets of C({\omega}^{\omega} ^k).