Sections of convex bodies and splitting problem for selections

Abstract

Let F 1 : X → Y 1 and F 2 : X → Y 2 be any convex-valued lower semicontinuous mappings and let L : Y 1 ⊕ 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 ε-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.