Sections of convex bodies and splitting problem for selections

Let F-1 : X -> Y-1 and F-2: X -> Y-2 be any convex-valued lower semicontinuous mappings and let L: Y-1 circle plus Y-2 -> Y be any linear surjection. The splitting problem is the problem of representation of any continuous selection f of the composite mapping L(F-1; F-2) in the form f = L(f(1); f(2)), where f(1) and f(2) are some continuous selections of F-1 and F-2, respectively. We prove that the splitting problem always admits an approximate solution with f(i) being an epsilon-selection (Theorem 2.1). We also propose a special case of finding exact splittings, whose occurrence is stable with respect to continuous variations of the data (Theorem 3.1) and we show that, in general, exact splittings do not exist even for the finite-dimensional range. (c) 2007 Elsevier Inc. All rights reserved.