On fixed point property under Lipschitz and uniform embeddings (Q1755684)

In [Stud. Math. 73, 225--251 (1982; Zbl 0506.46008)], \textit{S. Heinrich} and \textit{P. Mankiewicz} proved that every Lipschitz mapping from a separable \ Banach space to the dual of a separable space is almost everywhere weak$^*$ Gâteaux differentiable. In the article under review, the authors derive a local version of this result and give applications in fixed point theory. \par A point $x$ in a convex subset $C$ of a Banach space $X$ is a support point of $C$ if there exists a nonzero $x^*$ in the dual of $X$ such that $x^*(x) = \max\{x^*(y):y\in C\}$. The set $N(C)$ of non-support points of $C$ consists of the elements in $C$ that are not support points of $C$. The authors prove that, if $C$ is a closed convex subset of a separable Banach space with $N(C) \ne \emptyset$ and $Y$ is a separable Banach space, then a Lipschitz mapping $f:C\to Y^*$ is weak$^*$ Gâteaux differentiable almost everywhere (that is, off an Aronszajn null subset of $N(C)$). One consequence of this result is that, if $C$ is a nonempty closed bounded convex subset of a Banach space and $C$ Lipschitz embeds into a superreflexive space, then every isometry mapping $C$ into itself has a fixed point.