相关论文: Observations on quasihyperbolic geometry modeled o…
We investigate properties of quasihyperbolic balls and geodesics in Euclidean and Banach spaces. Our main result is that in uniformly smooth Banach spaces a quasihyperbolic ball of a convex domain is $C^1$-smooth. The question about the…
We consider the quasihyperbolic metric, and its generalizations in both the $n$-dimensional Euclidean space $R^n$, and in Banach spaces. Historical background, applications, and our recent work on convexity properties of these metrics are…
We study properties of quasihyperbolic geodesics on Banach spaces. For example, we show that in a strictly convex Banach space with the Radon-Nikodym property, the quasihyperbolic geodesics are unique. We also give an example of a convex…
We study convexity and starlikeness of quasihyperbolic and distance ratio metric balls on Banach spaces. In particular, problems related to these metrics on convex domains, and on punctured Banach spaces, are considered.
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…
An improved version of quasiinvariance property of the quasihyperbolic metric under M\"obius transformations of the unit ball in ${\mathbb R}^n, n \ge 2,$ is given. Next, a quasiinvariance property, sharp in a local sense, of the…
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…
Given a domain $G \subsetneq \Rn$ we study the quasihyperbolic and the distance ratio metrics of $G$ and their connection to the corresponding metrics of a subdomain $D \subset G$. In each case, distances in the subdomain are always larger…
Geometric properties of the quasi-hyperbolic Szekeres models are discussed and related to the quasi-spherical Szekeres models. Typical examples of shapes of various classes of 2-dimensional coordinate surfaces are shown in graphs; for the…
During the past thirty years hyperbolic type metrics have become popular tools also in modern mapping theory, e.g., in the study of quasiconformal and quasiregular maps in the euclidean $n$-space. We study here several metrics that one way…
A partially hyperbolic dynamical system is said to have the quasi-shadowing property if every pseudotrajectory can be shadowed by a sequence of points $(x_n)_{n\in \Z}$ such that $x_{n+1}$ is obtained from the image of $x_n$ by moving it by…
In three previous papers, we discussed quasidense multifunctions from a Banach space into its dual, or, equivalently, quasidense subsets of the product of a Banach space and its dual. In this paper, we survey (without proofs) some of the…
In this work we solve a couple of well known open problems related to the quasihyperbolic metric. In the case of planar domains, our first main result states that quasihyperbolic geodesics are unique in simply connected domains. As the…
We define metrics in space that are natural counterparts of the hyperbolic metric in plane domains, using the characterization of the hyperbolic metric due to Beardon and Pommerenke. We obtain inequalities for these metrics under…
In this paper, we introduce the concepts of short arc and length map in quasihyperbolic metric spaces, and obtain some geometric characterizations of Gromov hyperbolicity for quasihyperbolic metric spaces in terms of the properties of short…
We will consider inclusion of metric balls defined by the quasihyperbolic, the $j$-metric and the chordal metric. The quasihyperbolic metric and the $j$-metric are considered in general subdomains of $\mathbb{R}^n$ and in some particular…
We introduce the quasi-hyperbolicity constant of a metric space, a rough isometry invariant that measures how a metric space deviates from being Gromov hyperbolic. This number, for unbounded spaces, lies in the closed interval $[1,2]$. The…
The precise behavior of the quasi-hyperbolic metric near a $\mathcal C^{1,1}$-smooth part of the boundary of a domain in $\mathbb{R}^n$ is obtained.
We present and study a family of metrics on the space of compact subsets of $R^N$ (that we call ``shapes''). These metrics are ``geometric'', that is, they are independent of rotation and translation; and these metrics enjoy many…
Modifying the moduli of supporting convexity and supporting smoothness, we introduce new moduli for Banach spaces which occur, e.g., as lengths of catheti of right-angled triangles (defined via so-called quasi-orthogonality). These…