A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere
Authors:
Craig D. Hodgson, Igor Rivin and Warren D. Smith
Journal:
Bull. Amer. Math. Soc. 27 (1992), 246251
MSC (2000):
Primary 52B12; Secondary 51M10, 52A55, 68U05
MathSciNet review:
1149872
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Abstract: We describe a characterization of convex polyhedra in in terms of their dihedral angles, developed by Rivin. We also describe some geometric and combinatorial consequences of that theory. One of these consequences is a combinatorial characterization of convex polyhedra in all of whose vertices lie on the unit sphere. That resolves a problem posed by Jakob Steiner in 1832.
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 A. V. Aho, J. E. Hopcroft, and J. D. Ullman, The design and analysis of computer algorithms, AddisonWesley, Reading, MA, 1974. MR 0413592 (54:1706)
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 A. D. Aleksandrov, An application of the theorem of invariance of domain to existence proofs, Izv. Akad. Nauk SSSR Sci. Mat. 3 (1939), 243255. (Russian; English Summary)
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 [7]
 Alan F. Beardon, The geometry of discrete groups, SpringerVerlag, New York, 1983. MR 698777 (85d:22026)
 [8]
 A. L. Cauchy, Sur les polygones et polyèdres, 2nd memoir, J. École Polytech. 19 (1813), 8798.
 [9]
 H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved problems in geometry, SpringerVerlag, 1991. MR 1107516 (92c:52001)
 [10]
 M. Dillencourt and Warren D. Smith, Graphtheoretic aspects of inscribability, in preparation.
 [11]
 M. B. Dillencourt, Toughness and Delaunay triangulations, J. Discrete Comput. Geom. 5 (1990), 575601. MR 1067787 (91j:05068)
 [12]
 P. J. Federico, Descartes on polyhedra: A study of De Solidorum Elementis, Sources in the History of Mathematics and Physical Sciences, vol. 4, SpringerVerlag, New York, 1982. MR 680214 (84c:01024)
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 M. Grotschel, L. Lovasz, and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1 (1981), 169197. MR 625550 (84a:90044)
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 Branko Grünbaum, Convex polytopes, Wiley, New York, 1967. MR 0226496 (37:2085)
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 Barrett O'Neill, SemiRiemannian geometry; with applications to relativity, Academic Press, New York, 1983. MR 719023 (85f:53002)
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 Igor Rivin, On geometry of convex ideal polyhedra in hyperbolic 3space, Topology (to appear). MR 1204408 (94a:52025)
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 Igor Rivin, On geometry of convex polyhedra in hyperbolic 3space, PhD thesis, Princeton Univ., June 1986.
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 Igor Rivin, Intrinsic geometry of convex polyhedra in hyperbolic 3space, submitted.
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 Igor Rivin, Some applications of the hyperbolic volume formula of Lobachevsky and Milnor, submitted.
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 Igor Rivin and C. D. Hodgson, A characterization of compact convex polyhedra in hyperbolic 3space, Invent. Math. (to appear). MR 1193599 (93j:52015)
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 William P. Thurston, Geometry and topology of 3manifolds, Lecture notes, Princeton Univ., 1978.
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 Pravin M. Vaidya, A new algorithm for minimizing convex functions over convex sets, IEEE Sympos. Foundations of Computer Science, October 1989, pp. 338343.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S027309791992003038
PII:
S 02730979(1992)003038
Article copyright:
© Copyright 1992 American Mathematical Society
