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According to a result due to B.T. Polyak, a mapping between Hilbert spaces,which is $C^{1,1}$ around a regular point, carries a ball centered at thatpoint to a convex set, provided that the radius of the ball is small enough.The present paper considers the extension of such result to mappings defined ona certain subclass of uniformly convex Banach spaces. This enables one toextend to such setting a variational principle for constrained optimizationproblems, already observed in finite dimension, that establishes a convexbehaviour for proper localizations of them. Further variational consequencesare explored.

On the Polyak convexity principle and its application to variational analysis