Related papers: Hyperbolic width functions and characterizations o…
This paper contains a new concept to measure the width and thickness of a convex body in the hyperbolic plane. We compare the known concepts with the new one and prove some results on bodies of constant width, constant diameter and given…
Lipschitz constants for the width and diameter functions of a convex body in $\mathbb R^n$ are found in terms of its diameter and thickness (maximum and minimum of both functions). Also, a dual approach to thickness is proposed.
Extending Blaschke and Lebesgue's classical result in the Euclidean plane, it has been recently proved in spherical and the hyperbolic cases, as well, that Reuleaux triangles have the minimal area among convex domains of constant width $D$.…
Following Santal\'{o}'s approach, we prove several characterizations of a disc among bodies of constant width, constant projections lengths, or constant section lengths on given families of geodesics.
A strong consequence of quadratic forms becoming hyperbolic over the function field of a form is established. This result is invoked to obtain a new characterisation of hyperbolicity over function fields, and to recover a number of…
A definition of hyperbolic meromorphic functions is given and then we discuss the dynamical behavior and the thermodynamic formalism of hyperbolic functions on the Julia set. We prove the important expanding properties for hyperbolic…
We obtain new inequalities for certain hypergeometric functions. Using these inequalities, we deduce estimates for the hyperbolic metric and the induced distance function on a certain canonical hyperbolic plane domain.
In this paper our main aim is to develop some basic properties of hyperbolic valued metric spaces. We also establish the hyperbolic version of Banach contraction principle. Further we construct a hyperbolic valued metric on the space of all…
For every hyperplane $H$ supporting a convex body $C$ in the hyperbolic space $\mathbb{H}^d$ we define the width of $C$ determined by $H$ as the distance between $H$ and a most distant ultraparallel hyperplane supporting $C$. We prove that…
We define and study an extended hyperbolic space which contains the hyperbolic space and de Sitter space as subspaces and which is obtained as an analytic continuation of the hyperbolic space. The construction of the extended space gives…
For a hyperplane $H$ supporting a convex body $C$ in the hyperbolic space $\mathbb{H}^d$ we define the width of $C$ determined by $H$ as the distance between $H$ and a most distant ultraparallel hyperplane supporting $C$. The thickness…
We find complete hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of (elliptic) curvature functions which includes the higher order mean curvatures and their…
We consider the projectivization of Minkowski space with the analytic continuation of the hyperbolic metric and call this an extended hyperbolic space. We can measure the volume of a domain lying across the boundary of the hyperbolic space…
Although the hyperbolic metric possesses many remarkable properties, it is not defined on arbitrary subdomains of $\mathbb{R}^n$ with $n \geq 2$. This article introduces a new hyperbolic-type metric that provides an alternative approach to…
In this paper some concepts of convex analysis on hyperbolic space are studied. We first study properties of the intrinsic distance, for instance, we present the spectral decomposition of its Hessian. Next, we study the concept of convex…
In the recent paper "On a formula for sets of constant width in 2D", Comm. Pure Appl. Anal. 18 (2019), 2117-2131, we gave a constructive formula for all 2d sets of constant width. Based on this result we derive here a formula for the…
We give estimates on asymptotic dimensions of products of general hyperbolic spaces with following applications to the hyperbolic groups. We give examples of strict inequality in the product theorem for the asymptotic dimension in the class…
A classical problem in Complex Dynamics on hyperbolic domains is to characterize the hyperbolic step of parabolic functions. This topic has been studied by several authors, leading to different results and providing characterizations that…
We introduce continuum-wise hyperbolicity, a generalization of hyperbolicity with respect to the continuum theory. We discuss similarities and differences between topological hyperbolicity and continuum-wise hyperbolicity. A shadowing lemma…
In this paper we present a unique 4-dimensional body of constant width based on the classical notion of focal conics.