Roughness properties of norms on non-Asplund spaces (Q1191407)

Let \(X\) be a Banach space which is not an Asplund space, \(J\) the set- valued duality mapping on \(X\) and \(\varepsilon>0\). The authors prove that there is a (norm-) convergent sequence \((x_ n)\) in the unit sphere \(S_ 1(X)\) such that \(\text{dist}(J(x_ n),J(x_ m))\geq 1- \varepsilon\) for \(n\neq m\) and that there is a compact subset \(K\subset S_ 1(X)\) homeomorphic to the Cantor set and a selector \(\sigma\) of \(J\) such that \(\|\sigma(x)-\sigma(x')\|\geq 1-\varepsilon\) for different \(x,x'\in K\). Note that \(\sigma_ K(K)\) and \(\sigma(\{x_ n\})\) (\(\sigma\) any selector of \(J\) on \(\{x_ n\}\)) cannot be relatively compact and that the result holds for all equivalent norms. Klee had asked whether every Banach space can be renormed in order to have the following property: For every compact \(K\subset X\) there is a selector \(\sigma\) of \(J\) such that \(\sigma(K)\) is relatively compact. It follows that only Asplund spaces map have this property.