3 problems

Kadec renorming conjecture. Every sigma-fragmentable Banach space admits an equivalent Kadec renorming.

Renormability conjecture. If $K$ is a separable Rosenthal compact, then $\mathcal C(K)$ admits a locally uniformly convex renorming.

Let $X$ be a Banach space. A norm on $X$ is Gâteaux smooth when it is Gâteaux differentiable at every nonzero point. Lindenstrauss's conjecture. If $X$ admits a Gâteaux smooth norm…