Stability Constants of the Weak

Abstract. The main aim of the paper is to study some quantitative aspects of the stability of the weak fixed point property for nonexpansive maps in l1 (shortly, w-fpp). We focus on two complementary approaches to this topic. First, given a predual X of l1 such that the σ(l1, X)-fpp holds, we precisely establish how far, with respect to the Banach-Mazur distance, we can move from X without losing the w-fpp. The interesting point to note here is that our estimate depends only on the smallest radius of the ball in l1 containing all σ(l1, X)-cluster points of the extreme points of the unit ball. Second, we pass to consider the stability of the w-fpp in the restricted framework of preduals of l1. Namely, we show that every predual X of l1 with a distance from c0 strictly less than 3, induces a weak topology on l1 such that the σ(l1, X)-fpp holds.