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), 246-251

MSC (2000):
Primary 52B12; Secondary 51M10, 52A55, 68U05

MathSciNet review:
1149872

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[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. MR**0413592****[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.**[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. MR**2127379****[5]**E. M. Andreev,*Convex polyhedra in Lobačevskiĭ spaces*, Mat. Sb. (N.S.)**81 (123)**(1970), 445–478 (Russian). MR**0259734****[6]**E. M. Andreev,*Convex polyhedra of finite volume in Lobačevskiĭ space*, Mat. Sb. (N.S.)**83 (125)**(1970), 256–260 (Russian). MR**0273510****[7]**Alan F. Beardon,*The geometry of discrete groups*, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR**698777****[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. MR**1107516****[10]**M. Dillencourt and Warren D. Smith,*Graph-theoretic aspects of inscribability*, in preparation.**[11]**Michael B. Dillencourt,*Toughness and Delaunay triangulations*, Discrete Comput. Geom.**5**(1990), no. 6, 575–601. MR**1067787**, 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. MR**680214****[13]**M. Grötschel, L. Lovász, and A. Schrijver,*The ellipsoid method and its consequences in combinatorial optimization*, Combinatorica**1**(1981), no. 2, 169–197. MR**625550**, 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. MR**0226496****[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. MR**1184410****[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. MR**719023****[17]**Igor Rivin,*On geometry of convex ideal polyhedra in hyperbolic 3-space*, Topology**32**(1993), no. 1, 87–92. MR**1204408**, 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.**[19]**Igor Rivin,*Intrinsic geometry of convex polyhedra in hyperbolic*3-*space*, submitted.**[20]**Igor Rivin,*Some applications of the hyperbolic volume formula of Lobachevsky and Milnor*, submitted.**[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. MR**1193599**, 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. MR**1370757**, 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.**[24]**J. J. Stoker,*Geometrical problems concerning polyhedra in the large*, Comm. Pure Appl. Math.**21**(1968), 119–168. MR**0222765****[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.

Retrieve articles in *Bulletin of the American Mathematical Society*
with MSC (2000):
52B12,
51M10,
52A55,
68U05

Retrieve articles in all journals with MSC (2000): 52B12, 51M10, 52A55, 68U05

Additional Information

DOI:
https://doi.org/10.1090/S0273-0979-1992-00303-8

Article copyright:
© Copyright 1992
American Mathematical Society