Invariant strictly convex renormings

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.