## 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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## Additional Information

**Giulio Belletti**- Affiliation: Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
- ORCID: 0000-0002-7055-3335
- Email: giulio.belletti@sns.it
- Received by editor(s): February 18, 2020
- Received by editor(s) in revised form: May 21, 2020
- Published electronically: November 3, 2020
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 1125-1153 - MSC (2010): Primary 52B10
- DOI: https://doi.org/10.1090/tran/8215
- MathSciNet review: 4196389