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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Metrics $\rho$, quasimetrics $\rho ^s$ and pseudometrics $\inf \rho ^s$
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by K. V. Storozhuk PDF
Conform. Geom. Dyn. 21 (2017), 264-272 Request permission


Let $\rho$ be a metric on a space $X$ and let $s\geq 1$. The function $\rho ^s(a,b)=\rho (a,b)^s$ is a quasimetric (it need not satisfy the triangle inequality). The function $\inf \rho ^s(a,b)$ defined by the condition $\inf \rho ^s(a,b)=\inf \{\sum _0^n \rho ^s(z_i,z_{i+1})$ $z_0=a, z_n=b\}$ is a pseudometric (i.e., satisfies the triangle inequality but can be degenerate). We show how this degeneracy can be connected with the Hausdorff dimension of the space $(X,\rho )$. We also give some examples showing how the topology of the space $(X,\inf \rho ^s)$ can change as $s$ changes.
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Additional Information
  • K. V. Storozhuk
  • Affiliation: Sobolev Institute of Mathematics, 4 Koptyug Avenue, Novosibirsk, 630090, Russia – and – Novosibirsk State University, 2, Pirogova Street, Novosibirsk 630090, Russia – and – RUDN University, 6 Miklukho-Makiaya st, Moscow, Russia, 117198
  • MR Author ID: 637759
  • Email:
  • Received by editor(s): October 12, 2016
  • Received by editor(s) in revised form: April 17, 2017, May 7, 2017, and May 8, 2017
  • Published electronically: July 7, 2017
  • Additional Notes: This publication was supported by the Ministry of Education and Science of the Russian Federation (Agreement number 02.a03.21.0008).
  • © Copyright 2017 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 21 (2017), 264-272
  • MSC (2010): Primary 54E35; Secondary 28A78, 30C62
  • DOI:
  • MathSciNet review: 3668988