arXiv subjects

Zair Ibragimov

Publications and source records attributed to Zair Ibragimov.

3 recordsLinked to original sources

arXiv2017

Averaging one-point hyperbolic-type metrics

It is known that the $\tilde j$-metric, half-apollonian metric and scale-invariant Cassinian metric are not Gromov hyperbolic. These metrics are defined as a supremum of one-point metrics (i.e., metrics constructed using one boundary point) and the supremum is taken over all boundary points. The aim of this paper is to show that taking the average instead of the supremum yields a metric that preserves the Gromov hyperbolicity. Moreover, we show that the Gromov hyperbolicity constant of the resulting metric does not depend on the number of metrics used in taking the average. We also provide an example to show that the average of Gromov hyperbolic metrics is not, in general, Gromov hyperbolic.

math.MG↗

arXiv2015

Geometry of the Cassinian metric and its inner metric

The Cassinian metric and its inner metric have been studied for subdomains of the $n$-dimensional Euclidean space $\mathbb{R}^n$ ($n\ge 2$) by the first named author. In this paper we obtain various inequalities between the Cassinian metric and other related metrics in some specific subdomains of $\mathbb{R}^n$. Also, a sharp distortion property of the Cassinian metric under Möbius transformations of the unit ball is obtained.

math.MG↗

arXiv2014

Finite Ball Intersection Property of the Urysohn Universal Space

In a paper published posthumously, P.S. Urysohn constructed a complete, separable metric space that contains an isometric copy of every complete separable metric space, nowadays referred to as the Urysohn universal space. Here we study various convexity properties of the Urysohn universal space and show that it has a finite ball intersection property. We also note that Urysohn universal space is not hyperconvex.

math.MG↗