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Abstract. The Cassinian metric and its inner metric have been studied for subdomains of the n-dimensional Euclidean space R n (n ≥ 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 R n . Also, a sharp distortion property of the Cassinian metric under Möbius transformations of the unit ball is obtained.2010 Mathematics Subject Classification. 30C35, 30C20, 30F45, 51M10. Key words and phrases. Möbius transformation, the hyperbolic metric, the Cassinian metric, the distance ratio metric, the visual angle metric, the triangular ratio metric, inner metric.

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 Mo¨bius transformations. We give here a proof of Beardon's 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.

Abstract. 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.

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