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

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          Abstract

          We prove the theorem mentioned in the title, for \({\mathbb{R}}^n\), where \(n \ge 3\). 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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          Most cited references 8

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          Simplices of maximal volume in hyperbolic n-space

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            Geometry of Spaces of Constant Curvature

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              Volume increasing isometric deformations of convex polyhedra

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                Author and article information

                Journal
                2013-04-24
                2014-10-21
                10.1556/SSc.Math.2014.1292
                1304.6579

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                Custom metadata
                52B11, 52A38, 52A55
                42 pages, small corrections and additions, the final version
                math.DG math.MG

                Geometry & Topology

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