## 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

## Abstract:

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.## References

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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: stork@math.nsc.ru
- 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: https://doi.org/10.1090/ecgd/311
- MathSciNet review: 3668988