Invariant strictly convex renormings

TL;DR

This paper explores the existence and construction of strictly convex norms on Banach spaces that remain invariant under group actions, providing new tools and positive answers to longstanding questions.

Contribution

It introduces methods for constructing invariant strictly convex norms and resolves a specific open problem for the space c.

Findings

Existence of invariant strictly convex renormings for certain Banach spaces.

Development of tools for constructing such norms.

Positive answer to a question about c's renorming invariance.

Abstract

Motivated by the question of Mikael de la Salle, we investigate the problem of the existence of equivalent strictly convex norms on Banach spaces that are invariant with respect to an action of a group by linear isometries. We develop various tools for constructing such norms and prove several preservation results. We also answer positively a question of Antunes, Ferenczi, Grivaux and Rosendal whether there is a strictly convex renorming of $c$ invariant with respect to its full linear isometry group. Finally, we specialize to the spaces $L_1[0,1]$ and $C(K)$, where $K$ is compact Hausdorff, and indicate that amenability of groups plays a role in this problem.