Octahedrality and G\^ateaux smoothness

TL;DR

The paper demonstrates that Banach spaces with a Gâteaux smooth norm and a complemented  space can be renormed to be both Gâteaux smooth and octahedral, addressing a longstanding open problem.

Contribution

It provides a partial solution to a decades-old problem by showing the existence of such renormings under specific conditions.

Findings

Banach spaces with Gâteaux smooth norm and complemented  space can be renormed to be octahedral

The result advances understanding of the geometric structure of Banach spaces

Addresses a problem from the early 1990s

Abstract

We prove that every Banach space admitting a Gateaux smooth norm and containing a complemented copy of $\ell_1$ has an equivalent renorming which is simultaneously G\^ateaux smooth and octahedral. This is a partial solution to a problem from the early nineties.