Short proofs of Cambern's 1968 theorem and its generalizations applicable to fixed point theory

Abstract

First we provide short, elementary and self-contained proofs of all known results concerning the lower bounds of the Banach-Mazur distances between the space $c_0$ of sequences converging to $0$ and other $\ell_1$-preduals isomorphic to $c_0$. Then, we use our technique to obtain lower bounds for the Banach-Mazur distances between any two $\ell_1$-preduals $X$ and $Y$. Our estimate depends only on the smallest radiuses $r^*(X)$ and $r^*(Y)$ of the closed balls in $\ell_1$ containing, respectively, all $\sigma(\ell_1,X)$-cluster points and all $\sigma(\ell_1,Y)$-cluster points of the set of all extreme points of the closed unit ball in $\ell_1$, and for any values of $r^*(X)$ and $r^*(Y)$ it is sharp. We apply this result to show that for every $\ell_1$-predual $X$ with $r^*(X)< 1$, every $\ell_1$-predual $Y$ with the distance from $X$ strictly less than $\frac{3-r^*(X)}{1+r^*(X)}$ induces a weak$^*$ topology on $\ell_1$ such that $\ell_1$ has the $\sigma(\ell_1,Y)$-fixed point property for nonexpansive mappings. If we additionally assume that the standard basis in $\ell_1$ is $\sigma(\ell_1,X)$-convergent, then the estimate is precise. The same holds if the standard basis in $\ell_1$ has a finite number of $\sigma(\ell_1,X)$-cluster points and each of them has a finite number of non-zero coordinates. It should be emphasized that the value of this constant was known only for the space $c_0$ so far.