标题： 不变严格凸范数 显示英文标题

标题： Invariant strictly convex renormings

摘要： 受Mikael de la Salle的问题启发，我们研究了在Banach空间上是否存在等价的严格凸范数，这些范数对于由群通过线性等距作用的不变性。 我们开发了各种用于构造此类范数的工具，并证明了几种保持结果。 我们还积极回答了Antunes、Ferenczi、Grivaux和Rosendal的问题，即是否存在$c$的严格凸重范数，该范数对其完整的线性等距群是不变的。 最后，我们将重点放在空间$L_1[0,1]$和$C(K)$上，其中$K$是紧致豪斯多夫空间，并指出群的可约性在此问题中起着作用。 显示英文摘要

摘要： 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.