Lipschitz continuous parametrizations of set-valued maps with weakly convex images

Abstract

We continue the investigations started in [1]-[4], where weakly convex sets and set-valued maps with weakly convex images were studied. Sufficient conditions are found for the existence of a Lipschitz parametrization for a set-valued map with solidly smooth (generally, non-convex) images. It is also shown that the set-valued ε-projection on a weakly convex set and the unit outer normal vector to a solidly smooth set satisfy, as set functions, the Lipschitz condition and the Hölder condition with exponent 1/2, respectively, relative to the Hausdorff metric.