
Journal of Convex Analysis 08 (2001), No. 2, 409416 Copyright Heldermann Verlag 2001 When Can Points in Convex Sets be Separated by Affine Maps? Reinhard Börger Fachbereich Mathematik, Fernuniversität, 58084 Hagen, Germany For a class A of convex sets (in not necessarily finitedimensional) real vector spaces, let Sep A denote the class of all convex sets C such that the affine maps from C to elements of A separate points. If we restrict our attention to finitedimensional convex sets, there are only four possibilities for Sep_{f}A, denoting the intersection of Sep A and {C : C is a finitedimensional convex set}. Similarly, restriction to absolutely convex sets yields only three possibilities. In the general case, there are many possibilities for Sep A, at least as many as cardinals. In particular, there is no linefree convex set C such that for all linearly bounded convex sets D the affine maps from D to C separate points. Keywords: Absolutely convex set, absolutely affine map, linearly bounded convex set, linefree convex set. MSC: 52A01; 52A05, 04A40, 18A99 [ Fulltextpdf (249 KB)] 