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Compact hyperbolic tetrahedra with non-obtuse dihedral angles. (English) Zbl 1127.52010

Summary: Given a combinatorial description \(C\) of a polyhedron having \(E\) edges, the space of dihedral angles of all compact hyperbolic polyhedra that realize \(C\) is generally not a convex subset of \(\mathbb R^E\). If \(C\) has five or more faces, Andreev’s theorem states that the corresponding space of dihedral angles \(A_C\) obtained by restricting to non-obtuse angles is a convex polytope.
In this paper we explain why Andreev did not consider tetrahedra, the only polyhedra having fewer than five faces, by demonstrating that the space of dihedral angles of compact hyperbolic tetrahedra, after restricting to non-obtuse angles, is non-convex. Our proof provides a simple example of the “method of continuity”, the technique used in classification theorems on polyhedra by Alexandrow, Andreev, and Rivin-Hodgson.

MSC:

52B10 Three-dimensional polytopes
52A55 Spherical and hyperbolic convexity
51M09 Elementary problems in hyperbolic and elliptic geometries
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