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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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