  In the present paper, conditions under which the images of uniformly convex
sets through $C^{1,1}$ regular mappings between Banach spaces remain convex are
established. These conditions are expressed by a certain quantitative relation
betweeen the modulus of convexity of a given set and the global regularity
behaviour of the mapping on it. Such a result enables one to extend to a wide
subclass of convex sets the Polyak's convexity principle, which was originally
concerned with images of small balls around points of Hilbert spaces. In
particular, the crucial phenomenon of the preservation of convexity under
regular $C^{1,1}$ transformations is shown to include the class of $r$-convex
sets, where the value of $r$ depends on the regularity behaviour of the
involved transformation. Two consequences related to nonconvex optimization are
discussed: the first one is a sufficient condition for the global solution
existence for infinite-dimensional constrained extremum problems; the second
one provides a zero-order Lagrangian type characterization of optimality in
nonlinear mathematical programming.
