TITLE: What are some interesting ways of making new metrics out of old metrics?
QUESTION [9 upvotes]: If $d(x,y)$ and $e(x,y)$ are metrics then $d(x,y)+e(x,y)$ and $\frac{d(x,y)}{1+d(x,y)}$ are metrics.
If $d_i(x,y)$ for $i=1,\dots,n$ are metrics then so is $\sqrt{\sum_{i=1}^n{d_i^2(x,y)}}$
Are there other interesting ways of constructing new metrics from old metrics?

REPLY [2 votes]: Associated to any metric is a length structure. The length of an absolutely continuous path is the integral of it's metric derivative. Associated to any length structure is a (pseudo)-metric. If this is an honest metric, it is also a path metric: for any $x, y\in X$ and any $\varepsilon>0$, there is some $z\in X$ such that $d(x, z)+d(z, y)\le d(x, y)+\varepsilon$.
Taken from Gromov, "Metric structures for Riemannian and non-Riemannian spaces".