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In the present paper, the following convexity principle is proved: any closedconvex multifunction, which is metrically regular in a certain uniform sensenear 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 controlledLipschizian behaviour. This result, which is valid for mappings defined on asubclass of uniformly convex Banach spaces, can be regarded as a set-valuedgeneralization of the Polyak convexity principle. The latter, indeed, can bederived as a special case of the former. Such an extension of that principleenables one to build large classes of nonconvex multifunctions preserving theconvexity of small balls. Some applications of this phenomenon to the theory ofset-valued optimization are proposed and discussed.

Convexity of the images of small balls through perturbed convex multifunctions