Mazur intersection property for Asplund spaces

TL;DR

This paper demonstrates that, under certain set-theoretic assumptions, Asplund spaces of density  can be renormed to have the Mazur intersection property, showing the property is undecidable within ZFC.

Contribution

It proves the consistency of MIP normability for -density Asplund spaces with set-theoretic axioms, extending previous results and highlighting independence from ZFC.

Findings

MIP normability is consistent with ZFC under Martin's Maximum

Undecidability of MIP normability for -density Asplund spaces

Connection between set theory axioms and geometric properties of Banach spaces

Abstract

The main result of the present note states that it is consistent with the ZFC axioms of set theory (relying on Martin's Maximum MM axiom), that every Asplund space of density character $\omega_1$ has a renorming with the Mazur intersection property. Combined with the previous result of Jim\' enez and Moreno (based upon the work of Kunen under the continuum hypothesis) we obtain that the MIP normability of Asplund spaces of density $\omega_1$ is undecidable in ZFC.