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A characterization of Gram matrices of polytopes

Díaz Sánchez, Raquel (1999) A characterization of Gram matrices of polytopes. Discrete and Computational Geometry, 21 (4). pp. 581-601. ISSN 0179-5376

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Abstract

We give a characterization of the Gram matrices of spherical and finite-volume hyperbolic polytopes of a given combinatorial type. This is done in terms of the signs of certain minors of the Gram matrix.


Item Type:Article
Uncontrolled Keywords:n -dimensional polytopes
Subjects:Sciences > Mathematics > Geometry
ID Code:15709
References:

A.D. Alexandrow, Konvexe Polyeder, Berlin: Akademie Verlag, 1958 (original in Russian: GITTL, Moscow, 1950).

E.M. Andreev, On convex polyhedra in Lobachevskii spaces, Math. USSR-Sb. 10 (1970), 413–440.

E.M. Andreev, Convex polyhedra of finite volume in Lobachebskii space, Math. USSR-Sb. 12 (1970), 255–259.

E.M. Andreev, The intersection of the plane boundaries of a polytope with acute angles, Mat. Zametki 8 (1970), 521–527.

M.M. Bayer, C.W. Lee, Combinatorial aspects of convex polytopes, in Handbook of Convex Geometry, ed. by P.M. Gruber and J.M. Wills, Amsterdam: North-Holland, 1993.

J. Bokowski, B. Sturmfels, Polytopal and nonpolytopal spheres, an algorithmic approach, Israel J. Math., 57(3) (1987), 257–271.

A.L. Cauchy, Sur les polygones et poly`edres, J. Ecole Polytechnique 16 (1813), 87–99.

H.S.M. Coxeter, Discrete groups generated by reflections, Ann. of Math. 35(2) (1934), 588–621.

R. Díaz, Matrices de Gram y espacios de ángulos diédricos de poliedros, Ph.D. Thesis, Universidad Complutense de Madrid, June 1996.

B. Gr¨unbaum, Convex Polytopes, Pure and Applied Mathematics,Vol. XVI, NewYork: Interscience, 1967.

H.M. Hilden, M.T. Lozano, J.M. Montesinos, W.C. Whitten, On universal groups and threemanifolds, Invent. Math. 87 (1987), 441–456.

J.F.P. Hudson, Piecewise Linear Topology, New York: Benjamin, 1969.

B. Iversen, Hyperbolic Geometry, Cambridge: Cambridge University Press, 1992.

F. Lann´er, On complexes with transitive groups of automorphisms, Comm. Sem. Math. Univ. Lund 11 (1950), 1–71.

J. Milnor, The Schl¨afli differential equality, in Collected papers, Vol 1, Houston: Publish or Perish, 1994.

I. Rivin, C.D. Hodgson, A characterization of compact convex polyhedra in hyperbolic 3-space, Invent. Math. 111 (1993), 77–111.

I. Rivin, On geometry of convex ideal polyhedra in hyperbolic 3-space, Topology 32(1) (1993), 87–92.

I. Rivin, A characterization of ideal polyhedra in hyperbolic 3-space, Ann. of Math. 143(1) (1996), 51–70.

C.P. Rourke, B.J. Sanderson, Introduction to Piecewise-Linear Topology, New York: Springer-Verlag, 1972.

W. Thurston, The Geometry and Topology of 3-manifolds, Princeton University Notes, preprint.

E.B. Vinberg, Hyperbolic reflection groups, Russian Math. Surveys 40(1) (1985), 31–75.

E.B. Vinberg, Volumes of non-Euclidean polyhedra, Russian Math. Surveys 48(2) (1993), 15–45.

E.B. Vinberg (ed.), Geometry II, New York: Springer-Verlag, 1993.

Deposited On:21 Jun 2012 08:32
Last Modified:06 Feb 2014 10:30

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