Jakub Duda
On inverses of $\delta $-convex mappings

Comment.Math.Univ.Carolinae 42,2 (2001) 281-297.

Abstract:In the first part of this paper, we prove that in a sense the class of bi-Lipschitz $\delta $-convex mappings, whose inverses are locally $\delta $-convex, is stable under finite-dimensional $\delta $-convex perturbations. In the second part, we construct two $\delta $-convex mappings from $\ell _1$ onto $\ell _1$, which are both bi-Lipschitz and their inverses are nowhere locally $\delta $-convex. The second mapping, whose construction is more complicated, has an invertible strict derivative at $0$. These mappings show that for (locally) $\delta $-convex mappings an infinite-dimensional analogue of the finite-dimensional theorem about $\delta $-convexity of inverse mappings (proved in [7]) cannot hold in general (the case of $\ell _2$ is still open) and answer three questions posed in [7].

Keywords: delta-convex mappings, strict differentiability, normed linear spaces
AMS Subject Classification: Primary 47H99; Secondary 46G99, 58C20