Sections of convex bodies and splitting problem for selections

Let and be any convex-valued lower semicontinuous mappings and let be any linear surjection. The splitting problem is the problem of representation of any continuous selection f of the composite mapping L(F₁;F₂) in the form f=L(f₁;f₂), where f₁ and f₂ are some continuous selections of F₁ and F₂, respectively. We prove that the splitting problem always admits an approximate solution with fi 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.