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.
Last update: February 22, 2015.