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On inverses of $$\delta$$-convex mappings. (English) Zbl 1053.47522
Summary: 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 cannot hold in general (the case of $$\ell _2$$ is still open).

##### MSC:
 47H99 Nonlinear operators and their properties 46G99 Measures, integration, derivative, holomorphy (all involving infinite-dimensional spaces) 58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
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