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ISRN Geometry
Volume 2012 (2012), Article ID 484312, 13 pages
http://dx.doi.org/10.5402/2012/484312
Research Article

The Spherical Boundary and Volume Growth

1Department of Mathematics, National University of Ireland Maynooth, Maynooth, Kildare, Ireland
2National Survey and Cadastre of Denmark, Rentemestervej 8, 2400 Copenhagen, Denmark

Received 29 November 2011; Accepted 3 January 2012

Academic Editors: F. Balibrea, J. Montaldi, and A. Morozov

Copyright © 2012 Stephen M. Buckley and Simon L. Kokkendorff. 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 consider the spherical boundary, a conformal boundary using a special class of conformal distortions. We prove that certain bounds on volume growth of suitable metric measure spaces imply that the spherical boundary is “small” (in cardinality or dimension) and give examples to show that the reverse implications fail. We also show that the spherical boundary of an annular convex proper length space consists of a single point. This result applies to 𝑙2-products of length spaces, since we prove that a natural metric, generalizing such “norm-like” product metrics on a (possibly infinite) product of unbounded length spaces, is annular convex.