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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Acknowledgments The authors would like to thank Prof. V. D. Milman for suggesting the problem of studying order isomorphisms for cones, which led to the developments and results of this paper. The authors would also like to thank Prof. R. Schneider for his detailed comments regarding the written text. Both authors are supported by ISF grant No. 865/07.
- Order isomorphisms
- Ordered linear spaces