  We continue to investigate cases when the Repov\v{s}-Semenov splitting
problem for selections has an affirmative solution for continuous set-valued
mappings. We consider the situation in infinite-dimensional uniformly convex
Banach spaces. We use the notion of Polyak of uniform convexity and modulus of
uniform convexity for arbitrary convex sets (not necessary balls). We study
general geometric properties of uniformly convex sets. We also obtain an
affirmative solution of the splitting problem for selections of certain
set-valued mappings with uniformly convex images.
