  We consider a definition of a weakly convex set which is a generalization of
the notion of a weakly convex set in the sense of Vial and a proximally smooth
set in the sense of Clarke, from the case of the Hilbert space to a class of
Banach spaces with the modulus of convexity of the second order. Using the new
definition of the weakly convex set with the given modulus of nonconvexity we
prove a new retraction theorem and we obtain new results about continuity of
the intersection of two continuous set-valued mappings (one of which has
nonconvex images) and new affirmative solutions of the splitting problem for
selections. We also investigate relationship between the new definition and the
definition of a proximally smooth set and a smooth set.
