The almost fixed point property is not invariant under isometric renormings (Q1979350)

A closed convex subset \(C\) of a Banach space \((X, \|\cdot\|)\) has the \textit{almost fixed point property} (AFPP) if, for every nonexpansive mapping \(T:C\to C\), \(\inf\{ \|Tx - x\| : x\in C\} = 0\). If \(\mathcal{C}(X, \|\cdot\|)\) denotes the collection of closed convex subsets of \((X, \|\cdot\|)\) with the AFPP, \((X, \|\cdot\|)\) has \textit{stability of the} AFPP if, for every norm \(\|\cdot\|_1\) on \(X\) equivalent to \(\|\cdot\|\), \(\mathcal{C}(X, \|\cdot\|) = \mathcal{C}(X, \|\cdot\|_1)\). It was shown in [\textit{H. Fetter} et al., J. Math. Anal. Appl. 459, No. 2, 789--796 (2018; Zbl 1390.46012)] that a Banach space \(X\) is reflexive if and only if \(X\) has stability of the AFPP. In the article under review, the author defines a related notion of stability: A Banach space \((X, \|\cdot\|)\) has \textit{set-stability of the} AFPP \textit{under isometric renormings} if, for every norm \(|\cdot|\) on \(X\) such that \((X, \|\cdot\|)\) and \((X, |\cdot|)\) are isometric spaces, \(\mathcal{C}(X, \|\cdot\|) = \mathcal{C}(X, |\cdot|)\). By the previously mentioned result, every reflexive space satisfies this definition and the author asks if every Banach space has set-stability of the AFPP under isometric renormings. The author answers this question negatively by providing, among other examples, renormings of \(\ell_1\) and \(c_0\) that fail to have set-stability of the AFPP under isometric renormings.