Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2010, Article ID 191858, 15 pages doi:10.1155/2010/191858 Research Article Equivalence of the Apollonian and Its Inner Metric Peter H ¨ ast ¨ o, 1 S. Ponnusamy, 2 and S. K. Sahoo 2 1 Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014 Oulu, Finland 2 Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India Correspondence should be addressed to Peter H¨ ast¨ o, peter.hasto@helsinki.fi Received 29 November 2009; Accepted 22 March 2010 Academic Editor: Teodor Bulboac˘ a Copyright q 2010 Peter H ¨ ast¨ o et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We show that the equivalence of the Apollonian metric and its inner metric remains unchanged by the removal of a point from the domain. For this we need to assume that the complement of the domain is not contained in a hyperplane. This improves a result of the authors wherein the same conclusion was reached under the stronger assumption that the domain contains an exterior point. 1. Introduction and the Main Result The Apollonian metric was first introduced by Barbilian 1 in 1934-35 and then rediscovered by Beardon 2 in 1995. This metric has also been considered in 3–14. 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, 15–20; compare, for example, 21 for a historical overview and more references. One interesting historical point, made in 21, is that Barbilian himself proposed the name “Apollonian metric” in 1959, which was later independently coined by Beardon 2. Recently, the Apollonian metric has also been studied with certain group structures 22. In this paper we mainly study the equivalence of the Apollonian metric and its inner metric proving a result which is a generalization of Theorem 5.1 in 12. In addition, we also consider the j D metric and its inner metric, namely, the quasihyperbolic metric. Inequalities among these metrics see Table 1 and the geometric characterization of these inequalities in certain domains have been studied in 12, 13. We start by defining the above metrics and stating our main result. The notation used mostly is from the standard books by Beardon 23 and Vuorinen 24. We will be considering domains open connected nonempty sets D in the M¨ obius space R n : R n ∪ {∞}. The “Apollonian metric” is defined for x, y ∈ D  R n by