N. V. Abrosimov, E. Makai, jr., A. D. Mednykh, Yu. G. Nikonorov, and Günter Rote:

The infimum of the volumes of convex polytopes of any given facet areas is 0

Stud. Sci. Math. Hungarica 51 (2014), 466–519. doi:10.1556/SScMath.51.2014.4.1292, arXiv:1304.6579 [math.DG].

Abstract

We prove that, for any dimension n≥3, and any given sequence of f numbers forming the facet areas of an n-polytope with f facets, there are polytopes with the same facet areas and arbitrarily small volume. The case of the simplex was known previously. Also, the case n=2 was settled, but there the infimum was some well-defined function of the side lengths. For spherical and hyperbolic spaces, we give some necessary conditions for the existence of a convex polytope with given facet areas, and some partial results about sufficient conditions for the existence of (convex) tetrahedra.

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