Abstract.
Cyclic polytopes are characterized as simplicial polytopes satisfying Gale’s evenness condition (a combinatorial condition on facets relative to a fixed ordering of the vertices). Periodically-cyclic polytopes are polytopes for which certain subpolytopes are cyclic. Bisztriczky discovered a class of periodically-cyclic polytopes that also satisfy Gale’s evenness condition. The faces of these polytopes are braxtopes, a certain class of nonsimplicial polytopes studied by the authors. In this paper we prove that the periodically-cyclic Gale polytopes of Bisztriczky are exactly the polytopes that satisfy Gale’s evenness condition and are braxial (all faces are braxtopes). The existence of other periodically-cyclic Gale polytopes is open.
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Supported in part by a grant from the University of Kansas General Research Fund and by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.
Received: 20 September 2006
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Bayer, M.M., Bisztriczky, T. On Gale and braxial polytopes. Arch. Math. 89, 373–384 (2007). https://doi.org/10.1007/s00013-007-2163-x
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DOI: https://doi.org/10.1007/s00013-007-2163-x