Published in Rev. Roumanie Math. Pure Appl. 51(2006), no. 4, 433–452. INEQUALITIES AND GEOMETRY OF THE APOLLONIAN AND RELATED METRICS PETER H ¨ AST ¨ O, S. PONNUSAMY, AND S.K. SAHOO Abstract. In this paper we consider inequalities between the Apollonian metric, the Apol- lonian inner metric, the j G metric and the quasihyperbolic metric. We will show that many of these inequalities have nice geometric interpretations in terms of the domain G in which the metrics are defined. 1. INTRODUCTION In this paper we consider the Apollonian metric which was introduced in [1, 5]. Some basic features of this metric are that it is M¨obius invariant and equals the hyperbolic metric in balls and halfspaces. We also consider the inner metric of the Apollonian metric, the j G metric and its inner metric, the quasihyperbolic metric. We look at inequalities among these metrics and are especially interested in the geometric meaning of these inequalities. We start by defining the metrics and stating our main results. The notation used conforms largely to that of [4] and [34], the reader can consult Section 1.1 of this paper, if necessary. We will be considering domains (open connected non-empty sets) G in the M¨obius space R n . The Apollonian metric is defined for x, y ∈ G  R n by α G (x, y ) := sup a,b∈∂G log |a − x||b − y | |a − y ||b − x| (with the understanding that |∞ − x|/|∞ − y | = 1). This formula has a very nice geometric interpretation (indeed, this is one of the main reasons for the interest in the metric), see Section 1.2. It is in fact a metric if and only if the complement of G is not contained in a hyperplane and a pseudometric otherwise, as was noted in [5, Theorem 1.1]. This metric was introduced in [5] and has also been considered in [7, 12, 28, 30] and [15]–[24]. It should also be noted that the same metric has been studied from a different perspective under the name of the Barbilian metric for instance in [1, 2, 3, 6, 8, 26], cf. [9] for a historical overview and more references. One interesting historical point, made in [9], is that Barbilian himself suggested the name “Apollonian metric”, which was later independently coined by Beardon. 2000 Mathematics Subject Classification. Primary 30F45; Secondary 30C65. Key words and phrases. The Apollonian metric, the Barbilian metric, inner metrics, uniform domains, the comparison property, the quasihyperbolic metric. The first author was supported in part by the Academy of Finland, the Research Council of Norway, Project 160192/V30, and the National Board for Higher Mathematics (DAE, India). 1