Kolmogorov distance

The Kolmogorov distance is the largest vertical gap between two cumulative distribution functions:

It is an IPM whose test functions are indicator functions of half-lines $1\{x\le t\}$. It lies in $[0,1]$ and is the statistic of the Kolmogorov–Smirnov (KS) test, a classic non-parametric test for whether two samples come from the same distribution.

It is easy to compute and interpret, but it focuses on a single worst point and is mostly used for one-dimensional data.