Separation theorems for convex polytopes and finitely-generated cones derived from theorems of the alternative

We derive from Motzkin's Theorem that a point can be strongly separated by a hyperplane from a convex polytope and a finitely-generated convex cone. We state a similar result for Tucker's Theorem of the alternative. A generalisation of the residual existence theorem for linear equations which has recently been proved by Rohn [8] is a corollary. We state all the results in the setting of a general vector space over a linearly ordered (possibly skew) field. (C) 2011 Elsevier Inc. All rights reserved.