A Banach space is said to have Krein-Milman property (KMP in short) if every closed bounded convex set of it is a closed convex hull of its extreme points. Eg. Any reflexive space has KMP, $\ell_1$ has KMP.

A Banach space is said to have Radon-Nikodym property (RNP in short) if every closed bounded convex set has slices of arbitrary small diameter. Eg. Any reflexive space has RNP, $\ell_1$ has RNP, a dual separable Banach space has RNP. RNP has many other characterisations in terms of geometrical and also analytical.

It can be proved that that RNP implies KMP. Whether KMP implies RNP or not was not known for a long time. Is there any progress in recent past?

The equivalence between CPCP and strong regularity under Krein-Milman property, arXiv:2305.18976, by Ginés López-Pérez, Rubén Medina. $\endgroup$