Mazur’s criterion for continuity of convex set-valued maps

Abstract

The following result is proved: If a set-valued map F : D → cc (Y) is K-midconvex and for every function x : [0,1] → D there exists a Lebesgue measurable set-valued map G : [0,1] → c (Y) such that G(t) ⊂ F(x(t)) + K, t ϵ [0,1], then F is K-continuous. It is a generalization of some results obtained by I. Labuda and R.D. Mauldin (for additive functionals) and by R. Ger (for Jensen-convex functionals) related to a problem posed by S. Mazur.