Reflexivity is equivalent to stability of the almost fixed point property (Q1684791)

A closed convex subset \(C\) of a Banach space \((X,\|\cdot\|)\) has the almost fixed point property if, for every nonexpansive mapping \(T:C\to C\), \(\inf\{\,\|x - Tx\|:x\in C\} = 0\). In [Proc. Am. Math. Soc. 88, 44--46 (1983; Zbl 0523.47042)], \textit{S. Reich} characterized the closed convex subsets of reflexive Banach spaces with the almost fixed point property and in [Isr. J. Math. 71, No. 2, 211--223 (1990; Zbl 0754.47035)], \textit{I. Shafrir} showed that Reich's characterization does not extend to nonreflexive Banach spaces and gave a characterization of the almost fixed point property in general Banach spaces. In the paper under review, the authors consider the stability of the class of closed convex sets with the almost fixed point property under renormings of Banach spaces. In particular, they prove that, for every nonreflexive Banach space \((X,\|\cdot\|)\) and every \(\varepsilon>0\), there exists an equivalent norm \(|\cdot |\) on \(X\) whose Banach-Mazur distance from \(\|\cdot\|\) is less than \(1+\varepsilon\) and a closed convex subset \(C\) of \(X\) such \(C\) has the almost fixed point property for one of the norms \(\{\, \|\cdot\|, |\cdot | \}\), but fails the almost fixed point property for the other norm. As a consequence, the authors deduce that a Banach space is reflexive if and only if the family of closed convex subsets of the Banach space with the almost fixed point property is invariant under renormings.