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Upper bounds for edge-antipodal and subequilateral polytopes

Konrad J Swanepoel
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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 (Period. Math. Hungar. 38 (1999), 231--246). 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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Konrad J Swanepoel. 2006-01-26. Upper bounds for edge-antipodal and subequilateral polytopes. https://doi.org/10.1007/s-10998-007-1099-0

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