International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 191858, 15 pages
doi:10.1155/2010/191858
Research Article

Equivalence of the Apollonian and Its Inner Metric

Peter Hästö,1 S. Ponnusamy,2 and S. K. Sahoo2

1Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014 Oulu, Finland
2Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India

Received 29 November 2009; Accepted 22 March 2010

Academic Editor: Teodor Bulboacă

Copyright © 2010 Peter Hästö 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.

Abstract

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.