On linear selections of convex set-valued maps

Abstract

We study continuous subadditive set-valued maps taking points of a linear space X to convex compact subsets of a linear space Y. The subadditivity means that φ(x 1 + x 2) ⊂ φ(x 1) + φ(x 2). We characterize all pairs of locally convex spaces (X, Y) for which any such map has a linear selection, i.e., there exists a linear operator A: X → Y such that Ax ∈ φ(x), x ∈ X. The existence of linear selections for a class of subadditive maps generated by differences of a continuous function is proved. This result is applied to the Lipschitz stability problem for linear operators in Banach spaces.