Lexicographical separation in Rn

We prove a new separation theorem for any two subsets of Rⁿ with disjoint convex hulls, by linear operators, or isomorphisms, or isometries, in the sense of the lexicographical order of Rⁿ (or, equivalently, by “generalized half spaces”). Also, we prove that if one of the sets is the nonpositive orthant, then the isometry can be taken lexicographically nonnegative, or the isomorphism nonnegative (in the usual order). We give some applications.