Upper bounds for edge-antipodal and subequilateral polytopes

• Autores: Konrad J. Swanepoel
• Localización: Periodica mathematica hungarica, ISSN 0031-5303, Vol. 54, Nº. 1, 2007, págs. 99-106
• Idioma: inglés
• DOI: 10.1007/s-10998-007-1099-0
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• Resumen

  • A polytope in a finite-dimensional normed space is subequilateral if the length in the norm of each of its edges equals its diameter. Subequilateral polytopes occur in the study of two unrelated subjects: surface energy minimizing cones and edge-antipodal polytopes. We show that the number of vertices of a subequilateral polytope in any d-dimensional normed space is bounded above by (d / 2 + 1) d for any d = 2. The same upper bound then follows for the number of vertices of the edge-antipodal polytopes introduced by I. Talata [19]. This is a constructive improvement to the result of A. Pór (to appear) that for each dimension d there exists an upper bound f(d) for the number of vertices of an edge-antipodal d-polytopes. We also show that in d-dimensional Euclidean space the only subequilateral polytopes are equilateral simplices.