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delta-convex mappings; strict differentiability; normed linear spaces
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].
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