A variational principle for weighted Delaunay triangulations and hyperideal polyhedra. (English) Zbl 1181.52018

The problem of the existence of hyperbolic polyhedra with specified dihedral angles and combinatorial structure is not yet fully solved. I. Rivin [Ann. Math. (2) 143, No. 1, 51–70 (1996; Zbl 0874.52006)] characterized data of this type that correspond to ideal polyhedra (with all vertices on the sphere at infinity) and this was generalized by X. Bao and F. Bonahon [Bull. Soc. Math. Fr. 130, No. 3, 457–491 (2002; Zbl 1033.52009)] to hyperideal polyhedra (with all vertices on or beyond the sphere at infinity.)
An unpublished work of Thurston relates ideal polyhedra to Delaunay triangulations in the plane. As the author explains, weighted Delaunay triangulations (introduced by H. Edelsbrunner, Cambridge Monographs on Applied and Computational Mathematics. 7. Cambridge: Cambridge University Press. xii (2001; Zbl 0981.65028)] in the context of mesh generation) can be used in the same way to represent polyhedra with one ideal vertex and all other vertices hyperideal. The triangulations may be generalized to include cone points; the resulting structures are called “euclidean hyperideal circle patterns”. The main result of this paper characterizes when a particular set of combinatorial, intersection angle, and cone angle data can be realized by such a pattern, and hence by a hyperideal polyhedron of the type described. Volume formulae for various hyperideal polyhedra are also derived.


52B10 Three-dimensional polytopes
57R05 Triangulating
51M20 Polyhedra and polytopes; regular figures, division of spaces
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