This is an announcement for the paper "Weakly convex sets and modulus of nonconvexity" by Maxim V. Balashov and Dusan Repovs. Abstract: 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. Archive classification: math.FA math.GN Mathematics Subject Classification: 46A55, 52A01, 52A07, 54C60, 54C65 Citation: J. Math. Anal. Appl. 371:1 (2010), 113-127 Submitted from: dusan.repovs@guest.arnes.si The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1007.0162 or http://arXiv.org/abs/1007.0162