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Hodgson, Craig D.; Rivin, Igor; Smith, Warren D.

A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere. (English) Zbl 0759.52010

Bull. Am. Math. Soc., New Ser. 27, No. 2, 246-251 (1992).
From the abstract: “We describe a characterization of convex polyhedra in \(H^ 3\) in terms of their dihedral angles. We also describe some geometric and combinatorial consequences of that theory. One of these consequences is a combinatorial characterization of convex polyhedra in \(E^ 3\) all of whose vertices lie on the unit sphere.” Though the proofs are only sketched and refer mainly to until now unpublished bibliography, the paper is an interesting survey on the problem with an accurate list of classical and present references.
Reviewer: L.A.Santaló (Buenos Aires)

Cited in 2 Reviews
Cited in 29 Documents

MSC:

52A55 Spherical and hyperbolic convexity
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)

Keywords:

hyperbolic space; graphs; skeleton; Gauss-map; random access memory
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References:

[1] Alfred V. Aho, John E. Hopcroft, and Jeffrey D. Ullman, The design and analysis of computer algorithms, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Second printing; Addison-Wesley Series in Computer Science and Information Processing. · Zbl 0326.68005
[2] A. D. Aleksandrov, An application of the theorem of invariance of domain to existence proofs, Izv. Akad. Nauk SSSR Sci. Mat. 3 (1939), 243-255. (Russian; English Summary)
[3] A. D. Aleksandrov, The intrinsic metric of a convex surface in a space of constant curvature, Dokl. Acad. Sci. SSSR 45 (1944), 3-6. · Zbl 0061.37606
[4] A. D. Alexandrov, Convex polyhedra, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. Translated from the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze and A. B. Sossinsky; With comments and bibliography by V. A. Zalgaller and appendices by L. A. Shor and Yu. A. Volkov.
[5] E. M. Andreev, Convex polyhedra in Lobačevskiĭ spaces, Mat. Sb. (N.S.) 81 (123) (1970), 445 – 478 (Russian).
[6] E. M. Andreev, Convex polyhedra of finite volume in Lobačevskiĭ space, Mat. Sb. (N.S.) 83 (125) (1970), 256 – 260 (Russian).
[7] Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. · Zbl 0528.30001
[8] A. L. Cauchy, Sur les polygones et polyèdres, 2nd memoir, J. École Polytech. 19 (1813), 87-98.
[9] Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved problems in geometry, Problem Books in Mathematics, Springer-Verlag, New York, 1991. Unsolved Problems in Intuitive Mathematics, II. · Zbl 0748.52001
[10] M. Dillencourt and Warren D. Smith, Graph-theoretic aspects of inscribability, in preparation. · Zbl 0870.05048
[11] Michael B. Dillencourt, Toughness and Delaunay triangulations, Discrete Comput. Geom. 5 (1990), no. 6, 575 – 601. · Zbl 0727.05040 · doi:10.1007/BF02187810
[12] Pasquale Joseph Federico, Descartes on polyhedra, Sources in the History of Mathematics and Physical Sciences, vol. 4, Springer-Verlag, New York-Berlin, 1982. A study of the De solidorum elementis.
[13] M. Grötschel, L. Lovász, and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1 (1981), no. 2, 169 – 197. · Zbl 0492.90056 · doi:10.1007/BF02579273
[14] Branko Grünbaum, Convex polytopes, With the cooperation of Victor Klee, M. A. Perles and G. C. Shephard. Pure and Applied Mathematics, Vol. 16, Interscience Publishers John Wiley & Sons, Inc., New York, 1967. · Zbl 0152.20602
[15] Craig D. Hodgson, Deduction of Andreev’s theorem from Rivin’s characterization of convex hyperbolic polyhedra, Topology ’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 185 – 193. · Zbl 0765.52013
[16] Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. · Zbl 0531.53051
[17] Igor Rivin, On geometry of convex ideal polyhedra in hyperbolic 3-space, Topology 32 (1993), no. 1, 87 – 92. · Zbl 0784.52014 · doi:10.1016/0040-9383(93)90039-X
[18] Igor Rivin, On geometry of convex polyhedra in hyperbolic 3-space, PhD thesis, Princeton Univ., June 1986. · Zbl 0784.52014
[19] Igor Rivin, Intrinsic geometry of convex polyhedra in hyperbolic 3-space, submitted. · Zbl 0823.52008
[20] Igor Rivin, Some applications of the hyperbolic volume formula of Lobachevsky and Milnor, submitted. · Zbl 0823.52009
[21] Craig D. Hodgson and Igor Rivin, A characterization of compact convex polyhedra in hyperbolic 3-space, Invent. Math. 111 (1993), no. 1, 77 – 111. · Zbl 0784.52013 · doi:10.1007/BF01231281
[22] Igor Rivin, A characterization of ideal polyhedra in hyperbolic 3-space, Ann. of Math. (2) 143 (1996), no. 1, 51 – 70. · Zbl 0874.52006 · doi:10.2307/2118652
[23] Jakob Steiner, Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander, Reimer, Berlin, 1832; Appeared in J. Steiner’s Collected Works, 1881. · JFM 27.0418.11
[24] J. J. Stoker, Geometrical problems concerning polyhedra in the large, Comm. Pure Appl. Math. 21 (1968), 119 – 168. · Zbl 0159.24301 · doi:10.1002/cpa.3160210203
[25] William P. Thurston, Geometry and topology of 3-manifolds, Lecture notes, Princeton Univ., 1978.
[26] Pravin M. Vaidya, A new algorithm for minimizing convex functions over convex sets, IEEE Sympos. Foundations of Computer Science, October 1989, pp. 338-343. · Zbl 0852.90112
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