Short proofs of Cambern’s 1968 theorem and its generalizations applicable to fixed point theory - Japón, Maria; Malec, Marek; Piasecki, Łukasz (Topological Methods in Nonlinear Analysis, 2025)
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 _1-preduals isomorphic to c_0. Then, we use our technique to obtain lower bounds for the Banach-Mazur distances between any two _1-preduals X and Y. Our estimate depends only on the smallest radiuses r^*(X) and r^*(Y) of the closed balls in _1 containing, respectively, all σ(_1,X)-cluster points and all σ(_1,Y)-cluster points of the set of all extreme points of the closed unit ball in _1, and for any values of r^*(X) and r^*(Y) it is sharp. We apply this result to show that for every _1-predual X with r^*(X)< 1, every _1-predual Y with the distance from X strictly less than 3-r^*(X)1+r^*(X) induces a weak^* topology on _1 such that _1 has the σ(_1,Y)-fixed point property for nonexpansive mappings. If we additionally assume that the standard basis in _1 is σ(_1,X)-convergent, then the estimate is precise. The same holds if the standard basis in _1 has a finite number of σ(_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.