Zair Ibragimov

On the apollonian metric of domains in

We study the apollonian metric considered for sets in ℝ n by Beardon in 1995. This metric was first introduced for plane Jordan domains by Barbilian in 1934. For a special class of plane domains Beardon showed that conformal apollonian isometries are Möbius transformations. We give here a proof of Beardons result without conformality assumption. We show that the apollonian metric of a domain D is either conformal at every point of D, at only one point of D or at no point of D. We also present a suprising relation between convex bodies of constant width and the apollonian metric.

Conformality of the Apollonian Metric

The Apollonian metric aD of a domain % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

Apollonian isometries of planar domains are Möbius mappings

D\subset {\overline R}^{n}

Isometries of Relative Metrics

is rarely conformal. In fact, if it is conformal at one point then D is, up to a Möbius transformation, a complement of a convex body of constant width and if it is conformal at two points then D is a ball. We consider a quantity that measures the deviation of aD from being conformal. This quantity is essential in comparing the Apollonian metric to hyperbolic and quasihyperbolic metrics. We show that this quantity is invariant under Möbius transformations and compute it for some standard domains. We then use it to obtain sharp estimates between any two of the Apollonian, hyperbolic and quasihyperbolic metrics on such domains.

Hyperbolizing metric spaces

The Apollonian metric is a generalization of the hyperbolic metric, defined in a much larger class of open sets. Beardon introduced the metric in 1998, and asked whether its isometries are just the Möbius mappings. In this article we show that this is the case in all open subsets of the plane with at least three boundary points.

Geometry of the Cassinian Metric and Its Inner Metric

In this paper we consider isometries of relative metrics. We characterize isometries of the jD metric and of Seittenranta’s metric, as well as of their generalizations. We also derive some inequalities and results on the geodesics of these metrics.

The Third Symmetric Product of ℝ

It was proved by M. Bonk, J. Heinonen and P. Koskela that the quasihyperbolic metric hyperbolizes (in the sense of Gromov) uniform metric spaces. In this paper we introduce a new metric that hyperbolizes all locally compact noncomplete metric spaces. The metric is generic in the sense that (1) it can be defined on any metric space; (2) it preserves the quasiconformal geometry of the space; (3) it generalizes the j-metric, the hyperbolic cone metric and the hyperbolic metric of hyperspaces; and (4) it is quasi-isometric to the quasihyperbolic metric of uniform metric spaces. In particular, the Gromov hyperbolicity of these metrics also follows from that of our metric.

Convex Bodies of Constant Width and the Apollonian Metric

The Cassinian metric and its inner metric have been studied for subdomains of the n-dimensional Euclidean space

A scale-invariant Cassinian metric

{\mathbb {R}}^n