On the Monotonicity of the Volume of Hyperbolic Convex Polyhedra
نویسنده
چکیده
We give a proof of the monotonicity of the volume of nonobtuse-angled compact convex polyhedra in terms of their dihedral angles. More exactly we prove the following. Let P and Q be nonobtuse-angled compact convex polyhedra of the same simple combinatorial type in hyperbolic 3-space. If each (inner) dihedral angle of Q is at least as large as the corresponding (inner) dihedral angle of P , then the volume of P is at least as large as the volume of Q. Moreover, we extend this result to nonobtuse-angled hyperbolic simplices of any dimension. MSC 2000: 52A55, 52C29
منابع مشابه
Research Summary
In my doctoral dissertation (directed by W. P. Thurston) I studied the geometry of convex polyhedra in hyperbolic 3-space H3, and succeeded in producing a geometric characterization of dihedral angles of compact convex polyhedra by reducing the question to a convex isometric embedding problem in the De Sitter sphere, and resolving this problem. In particular, this produced a simple alternative ...
متن کاملHyperideal polyhedra in hyperbolic manifolds
Let (M, ∂M) be a 3-manifold with incompressible boundary that admits a convex co-compact hyperbolic metric (but is not a solid torus). We consider the hyperbolic metrics on M such that ∂M looks locally like a hyperideal polyhedron, and we characterize the possible dihedral angles. We find as special cases the results of Bao and Bonahon [BB02] on hyperideal polyhedra, and those of Rousset [Rou02...
متن کاملAn Inequality for Polyhedra and Ideal Triangulations of Cusped Hyperbolic 3-manifolds
It is not known whether every noncompact hyperbolic 3-manifold of finite volume admits a decomposition into ideal tetrahedra. We give a partial solution to this problem: Let M be a hyperbolic 3-manifold obtained by identifying the faces of n convex ideal polyhedra P1, . . . , Pn. If the faces of P1, . . . , Pn−1 are glued to Pn, then M can be decomposed into ideal tetrahedra by subdividing the ...
متن کاملOn some fixed points properties and convergence theorems for a Banach operator in hyperbolic spaces
In this paper, we prove some fixed points properties and demiclosedness principle for a Banach operator in uniformly convex hyperbolic spaces. We further propose an iterative scheme for approximating a fixed point of a Banach operator and establish some strong and $Delta$-convergence theorems for such operator in the frame work of uniformly convex hyperbolic spaces. The results obtained in this...
متن کاملRigidity of Circle Polyhedra in the 2-sphere and of Hyperideal Polyhedra in Hyperbolic 3-space
We generalize Cauchy’s celebrated theorem on the global rigidity of convex polyhedra in Euclidean 3-space E to the context of circle polyhedra in the 2-sphere S. We prove that any two convex and proper non-unitary c-polyhedra with Möbiuscongruent faces that are consistently oriented are Möbius-congruent. Our result implies the global rigidity of convex inversive distance circle packings in the ...
متن کامل