Right inverses of linear maps on convex sets
Let \(X\), \(Y\) be Banach spaces and \(L: X\to Y\) a continuous linear surjection. If \(M\) is a convex, non-open subset of \(X\), then the restriction of \(L\) to \(M\) generally fails to be an open map onto \(L(M)\) even when \(X\), \(Y\) are finite-dimensional and \(M\) has non-empty interior. Results connected to conditions guaranteeing that such a restriction has a continuous right inverse are presented. The cases when \(L(M)\) is a polyhedral body and \(M\) is bounded and \(X\), \(Y\) finite-dimensional and \(M\) a compact convex subset of \(X\) strictly convex along \(\text{Ker\,}L\) are particularly considered. It is deduced, as a corollary, the possibility of splitting a continuous function with values in the Minkowski sum of two convex bodies. The results are related to a problem posed by D. Repovš and P. Semenov.
scientific article; zbMATH DE number 5546602