Abstract
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.
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Communicated by Imre Bárány
This material is based upon work supported by the South African National Research Foundation under Grant number 2053752.
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Swanepoel, K.J. Upper bounds for edge-antipodal and subequilateral polytopes. Period Math Hung 54, 99–106 (2007). https://doi.org/10.1007/s-10998-007-1099-0
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DOI: https://doi.org/10.1007/s-10998-007-1099-0