Convex set functions ind-space

Given subsets A and B of Euclidean d-space R a and 0 ~ 0, we set A + B -{x + Y l x E A, y E B} and OA = {Ox Ix 6 A }. Further given a convex subset g2 of R d we shall say that a set function F : 2 ~ {~} ~ [0, + ~ ] is convex (concave} if the inequality F(OA + (1 O)B) ~ Or(A} + (1 0 ) / ' ( B ) (>=.) holds for all It ~ A, B c_ D, and all 0 < 0 < 1. Here we shall s tudy such set functions of the special form given in the following