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**A characterization of compact convex polyhedra in hyperbolic 3-space.**
*(English)*
Zbl 0784.52013

This paper is a comprehensive survey with detailed proofs based on the Ph. D. dissertation of the second author. A. D. Alexandrov, Convex Polyhedra (in Russian), Moscow, GITTL, 1950, (German translation: Berlin, Akademie Verlag 1958) has given a complete characterization of the shapes of compact convex polyhedra in the hyperbolic 3-space \(H^ 3\) in terms of the intrinsic hyperbolic metric on the boundary. The authors give an analogous extrinsic description in terms of their dihedral angles. The Gauss map of a compact convex polyhedron on the Euclidean 3-space \(E^ 3\) has been extended onto \(H^ 3\) by using the hyperboloid model of \(H^ 3\). The hyperbolic polar image \(G(P)\) of \(P\) has a realization as a “convex polyhedron” in the “de Sitter sphere” \(S^ 2_ 1\). The main result is

Theorem 1.1. A metric space \((M,g)\) homeomorphic to \(S^ 2\) can arise as the Gaussian image \(G(P)\) of a compact convex polyhedron \(P\) in \(H^ 3\) iff the following conditions hold:

(a) The metric \(g\) has constant curvature 1 away from a finite collection of cone points \(c_ i\).

(b) The cone angle at each \(c_ i\) is greater than \(2\pi\).

(c) The lengths of closed geodesics of \((M,g)\) are strictly greater than \(2\pi\).

These results have interesting consequences, e.g. Andreev’s characterization of compact convex polyhedra in \(H^ 3\) with dihedral angles not greater than \(\pi/2\) follows very easily [see the first author, Ohio State Univ. Math. Res. Inst. Publ. 1, 185-193 (1992; Zbl 0765.52013)]. Another result states that the face angles of a convex polyhedron \(P\) in \(H^ 3\) determine it up to congruence.

The second author applied the current approach to characterize convex polyhedra of finite volume just in the following review: [I. Rivin: On geometry of convex ideal polyhedra in hyperbolic 3-space, Topology 32, No. 1, 87-92 (1993; see the paper below)]. In the projective Cayley-Klein model of \(H^ 3\), an ideal convex hyperbolic polyhedron is represented by convex Euclidean polyhedron inscribed in the 2-sphere. So the result also characterizes the graphs which can be inscribed in the 2-sphere as the 1-skeleton of a convex Euclidean polyhedron. The question of the existence of such a description has been open since 1832, when it was raised by Jakob Steiner. As evidence that the above characterization is the desired one, Warren Smith has shown that Theorem 1 in the previous paper leads to a polynomial time algorithm for deciding whether a polyhedral graph can be inscribed in the sphere.

Theorem 1.1. A metric space \((M,g)\) homeomorphic to \(S^ 2\) can arise as the Gaussian image \(G(P)\) of a compact convex polyhedron \(P\) in \(H^ 3\) iff the following conditions hold:

(a) The metric \(g\) has constant curvature 1 away from a finite collection of cone points \(c_ i\).

(b) The cone angle at each \(c_ i\) is greater than \(2\pi\).

(c) The lengths of closed geodesics of \((M,g)\) are strictly greater than \(2\pi\).

These results have interesting consequences, e.g. Andreev’s characterization of compact convex polyhedra in \(H^ 3\) with dihedral angles not greater than \(\pi/2\) follows very easily [see the first author, Ohio State Univ. Math. Res. Inst. Publ. 1, 185-193 (1992; Zbl 0765.52013)]. Another result states that the face angles of a convex polyhedron \(P\) in \(H^ 3\) determine it up to congruence.

The second author applied the current approach to characterize convex polyhedra of finite volume just in the following review: [I. Rivin: On geometry of convex ideal polyhedra in hyperbolic 3-space, Topology 32, No. 1, 87-92 (1993; see the paper below)]. In the projective Cayley-Klein model of \(H^ 3\), an ideal convex hyperbolic polyhedron is represented by convex Euclidean polyhedron inscribed in the 2-sphere. So the result also characterizes the graphs which can be inscribed in the 2-sphere as the 1-skeleton of a convex Euclidean polyhedron. The question of the existence of such a description has been open since 1832, when it was raised by Jakob Steiner. As evidence that the above characterization is the desired one, Warren Smith has shown that Theorem 1 in the previous paper leads to a polynomial time algorithm for deciding whether a polyhedral graph can be inscribed in the sphere.

Reviewer: E.Molnár (Budapest)

### MSC:

52B10 | Three-dimensional polytopes |

52A55 | Spherical and hyperbolic convexity |

51M10 | Hyperbolic and elliptic geometries (general) and generalizations |

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\textit{C. D. Hodgson} and \textit{I. Rivin}, Invent. Math. 111, No. 1, 77--111 (1993; Zbl 0784.52013)

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