Right Inverses of linear maps on convex sets

We study, via continuous selections of multivalued maps, the problem of finding a right inverse to the restriction of a linear map to a convex body. Let $M$ be a convex body in a finite dimensional real space $X$ and let $L:X\rightarrow Y$ a (continuous) linear surjection, where $Y$ is also a real vector space. We address here the problem of finding a continuous function $s:L(M)\rightarrow M$ such that $L(s(x))=x$ for every $x\in L(M)$ (i.e. a right inverse of the restriction $L:M\rightarrow L(M)$). This has interesting applications: for example, when $M=A\times B$ with $A$ and $B$ convex body in $\R^n$ and $L$ the sum in $\R^n$, can we find $s=s_1,s_2:A+B\rightarrow A,B$ with $s_1(x)+s_2(x)=x$? \\We try to partially solve these kind of problem using convex closed multivalued mappings and in particular Michael theorem.