Metrics $\rho$, quasimetrics $\rho ^s$ and pseudometrics $\inf \rho ^s$

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.