Convexity of the images of small balls through nonconvex multifunctions

In the present paper, the following convexity principle is proved: any closed convex multifunction, which is metrically regular in a certain uniform sense near a given point, carries small balls centred at that point to convex sets, even if it is perturbed by adding C1,1 smooth mappings with controlled Lipschitzian behaviour. This result, which is valid for mappings defined on a subclass of uniformly convex Banach spaces, can be regarded as a set-valued generalization of the Polyak convexity principle. The latter, indeed, can be derived as a special case of the former. Such an extension of that principle enables one to build large classes of nonconvex multifunctions preserving the convexity of small balls. Some applications of this phenomenon to the theory of set-valued optimization are then proposed and discussed.