Faces of convex sets and Minkowski additive selections Original Research Article

A function S:varkappan → varkappan defined on the family varkappan of all compact, convex, and nonempty sets in imagen is said to be a face mapping if it is additive, and if for every A, S(A) is its face. It is known that if S assumes as its values only 0-dimensional faces, then there exists a lexicographical order on imagen such that for every A, a unique member of S(A) is the greatest element of A with respect to this order. We show that an extension of this result remains valid for all face mappings. We give also new results on additive selections.