Convexity properties of quasihyperbolic balls on Banach spaces

On Quasihyperbolic Geodesics in Banach Spaces

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 domain $Ω$ in a Banach space such that there is no geodesic between any given pair of points $x, y \in Ω\,.$ In addition, we prove that if $\mathrm{X}$ is a uniformly convex Banach space and its modulus of convexity is of a power type, then every geodesic of the quasihyperbolic metric, defined on a proper subdomain of $\mathrm{X}$, is smooth.

Quasihyperbolic Geometry in Euclidean and Banach Spaces

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 discussed.

Observations on quasihyperbolic geometry modeled on Banach spaces

In this paper, we continue our study of quasihyperbolic metric in Banach spaces. The main results of the paper present a criterion for smoothness of geodesics of quasihyperbolic type metrics in Banach spaces, under a Dini type condition on the weight function, which improves an earlier result of the two first authors. We also answer to a question posed by the two first authors in an earlier paper with R. Klén, and present results related to the question on smoothness of quasihyperbolic balls.

On smoothness of quasihyperbolic balls

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 smoothness of quasihyperbolic balls is old, originating back to the discussions of F.W. Gehring and M. Vuorinen in 1970's. To our belief, the result is new also in the Euclidean setting. We also address some other issues involving the smoothness of quasihyperbolic balls. We introduce an interesting application of quasihyperbolic metrics to renormings of Banach …

拟双曲域是CAT(2)的

Quasihyperbolic domains are CAT(2)

本文证明欧几里得空间真子域在拟双曲度量下是CAT(2)的，从而在所有维度上解决了Väisälä关于拟双曲测地线的多个猜想，并回答了Gehring-Vuorinen及Martio-Väisälä关于球面正则性和凸性的问题。

On removability properties of $ψ$-uniform domains in Banach spaces

Suppose that $E$ denotes a real Banach space with the dimension at least 2. The main aim of this paper is to show that a domain $D$ in $E$ is a $ψ$-uniform domain if and only if $D\backslash P$ is a $ψ_1$-uniform domain, and $D$ is a uniform domain if and only if $D\backslash P$ also is a uniform domain, whenever $P$ is a closed countable subset of $D$ satisfying a quasihyperbolic separation condition. This condition requires that the quasihyperbolic distance (w.r.t. $D$) between each pair of distinct points in $P$ has a lower bound greater than or equal to $\frac{1}{2}$.