The maximum volume of hyperbolic polyhedra

Belletti, Giulio

doi:10.1090/tran/8215

The maximum volume of hyperbolic polyhedra
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by Giulio Belletti PDF

Trans. Amer. Math. Soc. 374 (2021), 1125-1153 Request permission

Abstract:

We study the supremum of the volume of hyperbolic polyhedra with some fixed combinatorics and with vertices of any kind (real, ideal, or hyperideal). We find that the supremum is always equal to the volume of the rectification of the $1$-skeleton.

The theorem is proved by applying a sort of volume-increasing flow to any hyperbolic polyhedron. Singularities may arise in the flow because some strata of the polyhedron may degenerate to lower-dimensional objects; when this occurs, we need to study carefully the combinatorics of the resulting polyhedron and continue with the flow, until eventually we get a rectified polyhedron.

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