On norm attaining Lipschitz maps between Banach spaces (Q2821631)

In Proposition 4 of his famous paper [Isr. J. Math. 1, 139--148 (1963; Zbl 0127.06704)], \textit{J. Lindenstrauss} demonstrated that, if \(Y\) is a strictly convex Banach space, then every linear bounded operator \(T:c_0 \to Y\) that attains its norm is of finite rank. From this follows that, if \(Y\) is a strictly convex space isomorphic to \(c_0\), then the set of norm-attaining operators between \(c_0\) and \(Y\) does not contain any isomorphism, and consequently is not dense in the space of all linear bounded operators from \(c_0\) to \(Y\). The purpose of the paper under review is to find a nonlinear counterpart for the above result.NEWLINENEWLINELet \(X, Y\) be Banach spaces. For a Lipschitz map \(f: X \to Y\), denote by \(\mathrm{Lip}(f)\) the Lipschitz norm of \(f\), i.e., the best Lipschitz constant of \(f\). According to the author's definition, \(f\) attains its norm in a direction \(y \in Y\) if \(\|y\| =\mathrm{Lip}(f)\) and there is a sequence \((u_n, v_n)\) of distinct points of \(X\) such that NEWLINE\[NEWLINE \lim_n(f(u_n) - f(v_n))/\|u_n - v_n\| = y. NEWLINE\]NEWLINE It is demonstrated that if \(X\) is an asymptotically uniformly smooth separable Banach space and \(f\) is a Lipschitz isomorphism between \(X\) and \(Y\) which attains its Lipschitz norm in a direction \(y \in Y\), then the modulus of asymptotic smoothness of \(Y\) at \(y/\|y\|\) is controlled from above by the corresponding modulus of \(X\). This implies for instance that if \(X\) is a subspace of \(c_0\) and \(Y\) has the Kadec-Klee property (i.e., the weak and the norm topologies agree on the unit sphere of \(Y\)), then a Lipschitz isomorphism between \(X\) and \(Y\) cannot attain its norm in any direction \(y \in Y\).NEWLINENEWLINEThe work is dedicated to the memory of Joram Lindenstrauss (see the commemorative article [\textit{W. B. Johnson} (ed.) et al., Notices Am. Math. Soc. 62, No. 1, 26--37 (2015; Zbl 1338.01046)]).