Combinatorial optimization in geometry

  title={Combinatorial optimization in geometry},
  author={Igor Rivin},
  journal={Adv. Appl. Math.},
  • Igor Rivin
  • Published 6 July 1999
  • Mathematics
  • Adv. Appl. Math.
On an Elementary Proof of Rivin's Characterization of Convex Ideal Hyperbolic Polyhedra by their Dihedral Angles
In 1832, Jakob Steiner (Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einander, Reimer (Berlin)) asked for a characterization of those planar graphs which are combinatorially
1. Geometry of polyhedra and related subjects
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
Circle Patterns on Singular Surfaces
The possible intersection angles and singular curvatures of those circle patterns on Euclidean or hyperbolic surfaces with cone singularities ofHyperideal circle patterns are described, related to results on the dihedral angles of ideal or hyperidealhyperbolic polyhedra.
Rigidity of Thin Disk Configurations.
The main result of this thesis is a rigidity theorem for configurations of closed disks in the plane. More precisely, fix two collections C and C̃ of closed disks, sharing a contact graph which
Variational principles for circle patterns
A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there
From angled triangulations to hyperbolic structures
This survey paper contains an elementary exposition of Casson and Rivin's technique for finding the hyperbolic metric on a 3-manifold M with toroidal boundary. We also survey a number of applications
Many projectively unique polytopes
We construct an infinite family of 4-polytopes whose realization spaces have dimension smaller or equal to $$96$$96. This in particular settles a problem going back to Legendre and Steinitz: whether
Universality Theorems for Inscribed Polytopes and Delaunay Triangulations
It is proved that every primary basic semi-algebraic set is homotopy equivalent to the set of inscribed realizations (up to Möbius transformation) of a polytope, and that all algebraic extensions of Q are needed to coordinatize inscribed polytopes.
Characterizations of circle patterns and finite convex polyhedra in hyperbolic 3-space
The aim of this paper is to study finite convex polyhedra in three dimensional hyperbolic space $${\mathbb {H}}^3$$H3. We characterize the quasiconformal deformation space of each finite convex
Gauss images of hyperbolic cusps with convex polyhedral boundary
We prove that a 3-dimensional hyperbolic cusp with convex poly- hedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of


A characterization of ideal polyhedra in hyperbolic $3$-space
The goals of this paper are to provide a characterization of dihedral angles of convex ideal (those with all vertices on the sphere at infinity) polyhedra in H3, and also of those convex polyhedra
The virtual cohomological dimension of the mapping class group of an orientable surface
Let F = F ~ r be the mapping class group of a surface F of genus g with s punctures and r boundary components. The purpose of this paper is to establish cohomology properties of F parallel to those
The decorated Teichmüller space of punctured surfaces
A principal ℝ+5-bundle over the usual Teichmüller space of ans times punctured surface is introduced. The bundle is mapping class group equivariant and admits an invariant foliation. Several
Polyhedra of Small Order and Their Hamiltonian Properties
The smallest non-Hamiltonian planar graphs satisfying certain toughness-like properties are presented here, as are the smallest non -Hamiltonian, 3-connected, Delaunay tessellations and triangulations.
A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere
We describe a characterization of convex polyhedra in H 3 in terms of their dihedral angles, developed by Rivin. We also describe some geometric and combinatorial consequences of that theory. One of
A characterization of compact convex polyhedra in hyperbolic 3-space
SummaryIn this paper we study the extrinsic geometry of convex polyhedral surfaces in three-dimensional hyperbolic spaceH3. We obtain a number of new uniqueness results, and also obtain a
Improved Time Bounds for the Maximum Flow Problem
Possible improvements to the Ahuja-Orlin algorithm are explored and it is shown that the use of dynamic trees in the latter algorithm reduces the running time to $O(nm\log (({n / m})(\log U)^{{1 / 2}} + 2))$.
Les surfaces euclidiennes à singularités coniques. (Euclidean surfaces with cone singularities).
We prove in this paper that evry compact Riemann surface carries an euclidean (flat) conformal metric with precribed conical singularities of given angles, provided the Gauss-Bonnet relation is
The paper contains a complete description of the polyhedra of finite volume with dihedral angles not exceeding 90° in three-dimensional Lobacevskiĭ space. Bibliography: One item.