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  • 1 Sobolev Institute of Mathematics Novosibirsk Acad. Koptyug Av. 4 630090 Russia
  • 2 Novosibirsk State University Novosibirsk Pirogov St. 2 630090 Russia
  • 3 Hungarian Academy of Sciences A. Rényi Institute of Mathematics H-1364 Budapest Pf. 127 Hungary
  • 4 South Mathematical Institute of the V. S. C. of the Russian Academy of Sciences Vladikavkaz Markus Str. 22 362027 Russia
  • 5 Freie Universität Berlin Institut für Informatik Takustr. 9 14159 Berlin Germany
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We prove the theorem mentioned in the title for ℝn where n ≧ 3. 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. We also consider the cases of spherical and hyperbolic n-spaces. There 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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