The paper sets new limits on hyperbolic polyhedra volumes.
problem Finding upper bounds on volumes of hyperbolic polyhedra.
method Analyzes three types of polyhedra: ideal, compact with finite vertices, and finite volume with mixed vertices.
result Establishes new upper bounds for polyhedra volumes in hyperbolic space.
New bounds found for vertices of hyperbolic polyhedra in dimensions 5 to 12.
problem Determining minimum number of ideal and finite vertices in hyperbolic polyhedra.
method Geometric method of orthogonal gluings combined with double counting and recurrence relations.
result Improved lower bounds for vertices in all dimensions up to 12.
Improved volume estimates for right-angled polyhedra in hyperbolic space.
problem Estimating volumes of right-angled polyhedra in hyperbolic space.
method Combining Andreev theorem and Atkinson's results, improved upper volume estimates.
result Upper volume estimates for both compact and ideal right-angled polyhedra improved.
We consider a natural question: "Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence?" and a strongly related problem of K. Borsuk (1967): "Is it true that two ANR's homotopy dominating each other have the same homotopy type?" The answer was earlier known to be positive for ma…
We define the injectivity radius of a Coxeter polyhedron in H^3 to be half the shortest translation length among hyperbolic/loxodromic elements in the orientation-preserving reflection group. We show that, for finite-volume polyhedra, this number is always less than 2.6339..., and for compact polyhedra it is always les…
New proof shows integer norms on integer lattices are polyhedra.
problem Characterizing norms on integer lattices.
method New proof of Thurston's theorem.
result Unit ball of integer norms on integer lattices is a polyhedron.
As was pointed out by Nikulin [8] and Vinberg [10], a right-angled polyhedron of finite volume in hyperbolic n-space Hn has at least one cusp for n≥5. We obtain non-trivial lower bounds on the number of cusps of such polyhedra. For example, right-angled polyhedra of finite volume must have at least th…
Study improves learning algorithms for convex polyhedra in Hilbert spaces.
problem Learning convex polyhedra in Hilbert spaces.
method Proposes an algorithm for learning a polyhedron in a Hilbert space.
result Correctly classifies at least 1-ε of the distribution with high probability.
Study on mapping non-triangulable manifolds to finite polyhedra.
problem Mapping non-triangulable manifolds to finite polyhedra.
method Investigates the Alexandroff-Borsuk problem in non-triangulable manifolds.
result Existence of an ε-map inducing homotopy equivalence.
Paper solves long-standing problem of infinite ideal polyhedra in hyperbolic space.
problem Characterize infinite ideal polyhedra in hyperbolic 3-space.
method Introduced combinatorial Ricci flow for infinite ideal circle patterns.
result Proved characterization of infinite ideal circle patterns under specific conditions.
Intermediate logic of all convex polyhedra is axiomatized.
problem Defining and axiomatizing intermediate logic for convex polyhedra.
method Using Jankov-Fine formulas, classical polyhedral geometry, and p-morphic images to establish completeness.
result A finite axiomatisation of PL for all convex polyhedra.
Hyperbolic 3-manifolds tessellated by polyhedra embed geodesically in 4D hyperbolic space.
problem Embedding hyperbolic 3-manifolds in higher-dimensional hyperbolic spaces.
method Tessellation with right-angled polyhedra, proving embedding geodesically.
result Complete finite-volume hyperbolic 4-manifolds can be constructed from tessellated 3-manifolds.
Hexagonal triangulation remains rigid under certain conditions.
problem Maintaining the rigidity of hexagonal triangulation under specific transformations.
method PL conformality for hexagonal Delaunay triangulation.
result Rigidity theorem for convex ideal hyperbolic polyhedra.
Tarski's theorem on intuitionistic logic is extended to polyhedra.
problem Capturing topological dimension in intuitionistic logic.
method Introducing polyhedral lattices and proving their Heyting properties.
result Intuitionistic logic can detect topological dimension in polyhedra.
Study on hyperbolic polyhedra and their volume, proving finiteness of arithmetic groups.
problem Finiteness of arithmetic maximal reflection groups in hyperbolic polyhedra.
method Observation of volume distribution and recent work with M. Fraczyk and S. Hurtado.
result Proof of finiteness of arithmetic maximal reflection groups.
Unified theory connects local and global geometric properties of random hyperbolic polyhedra.
problem Understanding the conformal type of unimodular random infinite trivalent hyperbolic polyhedra.
method Developed a geometric and probabilistic theory using disk triangulations and circle patterns.
result Established a sharp dichotomy between parabolic and hyperbolic types based on local geometric characteristics.
The study shows products of certain polyhedra have a bounded index property.
problem Understanding the bounded index property for products of polyhedra.
method Analyzing sufficient conditions for the bounded index property in products of polyhedra.
result Products of closed Riemannian manifolds with negative sectional curvature have the bounded index property.
The integer hull of a polyhedron is the convex hull of the integer points contained in it. We show that the vertices of the integer hulls of a rational family of polyhedra of size O(n) have quasipolynomial coordinates. As a corollary, we show that the stable commutator length of elements in a surgery family is a ratio …
We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse of finite spaces induces a simplicial …
This work is motivated by two problems: 1) The approach of manifolds and spaces by triangulations. 2) The complexity growth in sequences of polyhedra. Considering both problems as related, new criteria and methods for approximating smooth manifolds are deduced. When the sequences of polyhedra are obtained by the action…
Upper bounds for volumes of hyperbolic polyhedra and links are derived.
problem Finding upper limits for volumes of generalized hyperbolic polyhedra and links.
method Application of Belletti's theorem and analysis of polyhedra with triangular faces and trivalent vertices.
result Improved upper bounds for volumes of hyperbolic polyhedra and links are derived.
The study examines the normalized volumes of right-angled hyperbolic polyhedra and their spectra.
problem Investigating the normalized volumes of right-angled hyperbolic polyhedra.
method Analyzing the sets of compact and ideal right-angled hyperbolic polyhedra to determine their normalized volume spectra.
result The spectra of normalized volumes for compact and ideal right-angled hyperbolic polyhedra have specific intervals and densities.
Uniform covers with a finite-dimensional nerve are rare (i.e., do not form a cofinal family) in many separable metric spaces of interest. To get hold on uniform homotopy properties of these spaces, a reasonably behaved notion of an infinite-dimensional metric polyhedron is needed; a specific list of desired properties …
Researchers explore valuations on polyhedra and topological arrangements without imposing algebraic structures.
problem Understanding valuations on polyhedra and their connections to topological arrangements.
method Generalizes the setting of valuations on convex polyhedra to collections of defining hyperplanes without imposing algebraic structures.
result Uncovered a close relationship between scissors congruence problems and finite hyperplane arrangements.
Löbell polyhedra have small systoles and are quasi-arithmetic.
problem Finding compact hyperbolic polyhedra with small systoles.
method Elementary and conceptual means to observe systole behavior, number theoretic invariants to refine results.
result Löbell polyhedra give examples of closed hyperbolic 3-manifolds with arbitrarily small systole and are quasi-arithmetic.
Introduces Coxeter polyhedra in various geometries.
problem None explicitly stated, focuses on introducing concepts.
method Introduction of Coxeter polyhedra in spherical, Euclidean, and hyperbolic geometries.
result Fundamental theorems and descriptions of polyhedra and tessellations.
New hyperbolic polyhedra with π/3 angles and volumes calculated.
problem Finding new hyperbolic polyhedra with specific dihedral angles.
method Constructed a new sequence of hyperbolic polyhedra with π/3 angles and determined their volumes. result Volumes of some constructed polyhedra determined.
Study of polyhedra on a sphere in projective 3-space.
problem Characterizing polyhedra with vertices on a sphere.
method Purely combinatorial and linear programming approaches.
result Characterization of dihedral angles and hyperbolic-de Sitter structure.
Characterizes infinite ideal polyhedra in hyperbolic 3-space and proves their existence and rigidity.
problem Characterize infinite ideal polyhedra in hyperbolic 3-space.
method Study ideal circle patterns (ICPs) and develop a uniform Ring Lemma via pointed Gromov-Hausdorff convergence.
result Establish existence and rigidity of embedded ICPs and infinite ideal polyhedra (IIP).
The paper extends Gelfand-Kapranov-Zelevinsky construction to hyperbolic Riemann surfaces with punctures.
problem Stratifying the space of weight vectors for hyperbolic Riemann surfaces with punctures.
method Analogous to Gelfand-Kapranov-Zelevinsky construction, associates polyhedral fans to hyperbolic Riemann surfaces with punctures.
result The secondary fan of a hyperbolic Riemann surface with punctures is the normal fan of a convex polyhedron, the secondary polyhedron.
Study mutant pairs of hyperbolic polyhedra, focusing on commensurability.
problem Determine commensurability of mutant pairs of hyperbolic polyhedra.
method Introduce mutation concept, develop new techniques for non-cusped polyhedra.
result New techniques needed for studying mutant pairs of polyhedra.
The study sets limits on dihedral angles of large hyperbolic polyhedra.
problem Establishing bounds on dihedral angles of hyperbolic Coxeter polyhedra.
method Developed a constructive procedure for Coxeter polyhedra with prescribed dihedral angles.
result Classification of ADEG-polyhedra with specific dihedral angles and no disjoint facets.
Algorithm finds smallest right-angled hyperbolic polyhedra.
problem Finding the smallest volume right-angled hyperbolic polyhedra.
method Algorithm implemented on computer using Orb to compute volumes.
result First 825 polyhedra in the list of smallest volume right-angled hyperbolic polyhedra in dimension 3 have been determined.
We prove that if the fundamental group of an orientable finite volume hyperbolic 3-manifold has finite index in the reflection group of a right-angled ideal polyhedra in H3 then it has a co-final tower of finite sheeted covers with positive rank gradient. The manifolds we provide are also known to have co-f…
Easy condition for local k-connectedness in inverse limits of polyhedra.
problem Local k-connectedness of inverse limits of polyhedra.
method Easy condition for local k-connectedness.
result Provided an easily verifiable condition.
The study finds upper bounds and computes volumes of ideal right-angled polyhedra in Lobachevsky space.
problem Finding upper bounds and computing volumes of ideal right-angled polyhedra in Lobachevsky space.
method Analyzing a class of right-angled polyhedra with vertices on the absolute, obtaining upper bounds on volumes, computing volumes for polyhedra with up to 23 faces, and introducing the class of polyhedra with isolated triangles.
result Minimum volumes are realized on antiprisms and twisted antiprisms, and the first 248 values of volumes are presented.
This is an investigation of the role of shuffling and concatenating in the theory of graph drawing. A simple syntactic description of these and related operations is proved complete in the context of finite partial orders, as general as possible. An explanation based on that is given for a previously investigated colla…
The article explores toric spaces of regular polyhedra, highlighting rational and non-rational cases.
problem Exploring toric spaces associated with regular convex polyhedra.
method Symplectic and complex toric spaces associated with five regular convex polyhedra.
result The regular dodecahedron and icosahedron cannot be treated via standard toric geometry.
We review several results related to the characterization of polyhedra in hyperbolic 3-space. In particular we present Rivin's theorem that gives a characterization of compact convex hyperbolic polyhedra, and Hodgson's proof of the Adreev's theorem. We also review the analogous characterization of ideal polyhedra, and …
Improved bounds on ideal vertices in right-angled hyperbolic polyhedra.
problem Finding bounds on ideal vertices in hyperbolic polyhedra.
method Improved Nikulin's inequality and Nonaka's lower bound.
result Shorter proofs and improved bounds on ideal vertices.
Study calculates mass of special polyhedra in hyperbolic space.
problem Evaluating mass in hyperbolic geometry.
method Used upper half space model and special polyhedra.
result Evaluated mass functional on polyhedra.
Mass in relativity linked to polyhedra geometry.
problem Mass in general relativity.
method Riemannian polyhedra geometry.
result Mass connected to polyhedra geometry.
Flexible models of non-flexible polyhedra explained.
problem Non-flexible Siamese dipyramids behave like flexible ones.
method Simple mathematical method to explain model flexibility.
result Physical models of Siamese dipyramids are flexible.
Polyhedra volume conjecture supports Stoker conjecture weakly.
problem Proving the Stoker conjecture for polyhedra.
method Using an extension of Montcouquiol and Weiss' result on polyhedra angles as local coordinates.
result Volume Conjecture for polyhedra implies a weak version of the Stoker conjecture.
This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angl…
Study approximates Riemannian manifolds using polyhedra.
problem Understanding Tullio Regge's approximation theorem.
method Proof of Regge theorem using polyhedra approximation.
result Integral of scalar curvature approximated by polyhedral curvature.
If X is a proper CAT(-1)-space and Γ a non-elementary discrete group of isometries acting properly discontinuously on X, it is shown that the geodesic flow on the quotient space Y=X/Γ is topologically mixing, provided that the generalized Busemann function has zeros on the boundary ∂X and the non-wanderin…
The main result is that every complete finite area hyperbolic metric on a sphere with punctures can be uniquely realized as the induced metric on the surface of a convex ideal polyhedron in hyperbolic 3-space. A number of other observations are included.