Order isomorphisms in cones and a characterization of duality for ellipsoids |
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Authors: | Shiri Artstein-Avidan Boaz A Slomka |
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Institution: | 1. School of Mathematical Science, Tel Aviv University, Ramat Aviv, 69978, Tel Aviv, Israel
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Abstract: | We study order isomorphisms in finite-dimensional ordered vector spaces. We generalize theorems of Alexandrov, Zeeman, and Rothaus (valid for ??non-angular?? cones) to wide classes of cones, including in particular polyhedral cones, using a different and novel geometric method. We arrive at the following result: whenever the cone has more than n generic extremal vectors, an order isomorphism must be affine. In the remaining case, of precisely n extremal rays, the transform has a restricted diagonal form. To this end, we prove and use a new version of the well-known Fundamental theorem of affine geometry. We then apply our results to the cone of positive semi-definite matrices and get a characterization of its order isomorphisms. As a consequence, the polarity mapping is, up to a linear map, the only order-reversing isomorphism for ellipsoids. |
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