Density of the set of renormings in \(c_{0}\) without asymptotically isometric copies of \(c_{0}\) and failing to have the fixed point property (Q2871098)

Given a Banach space \((X, \|\cdot\|)\), let \(\mathcal{P}(X)\) denote the set of equivalent norms on \(X\) endowed with the metric \(\rho(p, q) = \sup\{|p(x)-q(x)| : \| x\| \leq 1\}\) if \(p, q\in \mathcal{P}(X)\). \textit{T. Domínguez Benavides} [Arab. J. Math. 1, No. 4, 431--438 (2012; Zbl 1273.47083)] proved that the set of equivalent norms on \(c_0\) that fail to have the fixed point property for nonexpansive mappings is dense in \(\mathcal{P}(c_0)\). \textit{C. A. Hernández-Linares} et al. [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 77, 112--117 (2013; Zbl 1264.46005)] proved that the set of equivalent norms on \(c_0\) that fail to contain an asymptotically isometric copy of \(c_0\) is also dense in \(\mathcal{P}(c_0)\). The main result in the article under review unifies these two results. In particular, the authors prove that, if \(\|\cdot\|\) is an equivalent norm on \(c_0\) and \(\varepsilon>0\), there exists a norm \(|\!|\!|\cdot|\!|\!|\) in \(\mathcal{P}(c_0)\) such that \(\rho( \|\cdot\|, |\!|\!|\cdot|\!|\!| )< \varepsilon\), \((c_0, |\!|\!|\cdot|\!|\!|)\) fails to have the fixed point property for nonexpansive mappings, and \((c_0, |\!|\!|\cdot|\!|\!|)\) fails to contain an asymptotically isometric copy of \(c_0\).