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Journal of Convex Analysis 08 (2001), No. 2, 409--416
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 finite-dimensional) 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 finite-dimensional convex sets, there are only four possibilities for SepfA, denoting the intersection of Sep A and {C : C is a finite-dimensional 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 line-free 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, line-free convex set.

MSC: 52A01; 52A05, 04A40, 18A99

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