  Let $A+B$ be the pointwise (Minkowski) sum of two convex subsets $A$ and $B$
of a Banach space. Is it true that every continuous mapping $h:X \to A+B$
splits into a sum $h=f+g$ of continuous mappings $f:X \to A$ and $g:X \to B$?
We study this question within a wider framework of splitting techniques of
continuous selections. Existence of splittings is guaranteed by hereditary
invertibility of linear surjections between Banach spaces. Some affirmative and
negative results on such invertibility with respect to an appropriate class of
convex compacta are presented. As a corollary, a positive answer to the above
question is obtained for strictly convex finite-dimensional precompact spaces.
