A function S:K n → K n defined on the family K n of all compact, convex, and nonempty sets in R n 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 R n 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.