Convex polarities over ordered fields

We use tools and methods from real algebraic geometry (spaces of ultrafilters, elimination of quantifiers) to formulate a theory of convexity in K^N over an arbitrary ordered field. By defining certain ideal points (which can be viewed as generalizations of recession cones) we obtain a generalized notion of polar set. These satisfy a form of polar duality that applies to general convex sets and does not reduce to classical duality if K is the field of real numbers. As an application we give a partial classification of total orderings of Artinian local rings and two applications to ordinary convex geometry over the real numbers.