On Inverses of -convex Mappings

In the first part of this paper, we prove that in a sense the class of bi-Lipschitz δ-convex mappings, whose inverses are locally δ-convex, is stable under finite-dimensional δ-convex perturbations. In the second part, we construct two δ-convex mappings from l1 onto l1, which are both bi-Lipschitz and their inverses are nowhere locally δ-convex. The second mapping, whose construction is more complicated, has an invertible strict derivative at 0. These mappings show that for (locally) δ-convex mappings an infinitedimensional analogue of the finite-dimensional theorem about δ-convexity of inverse mappings (proved in [7]) cannot hold in general (the case of l2 is still open) and answer three questions posed in [7].