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<p>A convex polytope <i>P</i> is monotypic if every polytope with the same set of outer normal vectors to its facets is isomorphic to <i>P</i>. Such a polytope <i>P</i> is also characterized by the fact that each non-empty intersection of <i>P</i> with a translate is homothetic to a summand of <i>P</i>. A strongly monotypic polytope <i>P</i> is such that the facet normals determine the isomorphism class of the arrangement of the hyperplanes spanned by its facets. It was shown in an earlier paper on the topic that, if <i>P</i> is strongly monotypic, then each non-empty intersection of <i>P</i> with a translate is a summand of <i>P</i>. However, the converse was left open. A proof of this has recently been given by Vuong Bui; however, a striking feature of the alternative proof given here, which as in the original paper uses representations, is that it reduces the problem to the already known 3-dimensional case.</p>

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On Monotypic Polytopes

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月9日

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  • Peter McMullen

摘要

A convex polytope P is monotypic if every polytope with the same set of outer normal vectors to its facets is isomorphic to P. Such a polytope P is also characterized by the fact that each non-empty intersection of P with a translate is homothetic to a summand of P. A strongly monotypic polytope P is such that the facet normals determine the isomorphism class of the arrangement of the hyperplanes spanned by its facets. It was shown in an earlier paper on the topic that, if P is strongly monotypic, then each non-empty intersection of P with a translate is a summand of P. However, the converse was left open. A proof of this has recently been given by Vuong Bui; however, a striking feature of the alternative proof given here, which as in the original paper uses representations, is that it reduces the problem to the already known 3-dimensional case.