  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 centered at that point to convex sets,
even if it is perturbed by adding C^{1,1} smooth mappings with controlled
Lipschizian 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 proposed and discussed.
