# Whether Krein-Milman property implies Radon-Nikodym property

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?

• See also the introduction of The equivalence between CPCP and strong regularity under Krein-Milman property, arXiv:2305.18976, by Ginés López-Pérez, Rubén Medina. Sep 12, 2023 at 7:12

• I would like to add one comment and one question in the same thread. RNP is closely related to Asplund property. A Banach space $X$ is said to be an Asplund space if every continuous convex function is Frechet differentiable in a dense $G_\delta$ set. Again it has many equivalent characterizations like $X$ is Asplund if and only if every equivalent norm on it has at least one point of Frechet differentiability also, $X$ is Asplund iff $X^*$ have RNP. It is known that if $X$ has an equivalent norm which is Frechet diff then $X$ is Asplund. My question is whether the converse is also true? Nov 23, 2019 at 13:27
• For what is worth, these properties coincide also in Lipschitz-free spaces, but it is for stronger reason (failure of the RNP already implies the presence of $L_1$). It is proved in Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions. Trans. Amer. Math. Soc. 375 (2022), no. 5, 3529–3567. Apr 4, 2023 at 11:15