Let $F,G:\mathbb{R}\to[0,1]$ be two cumulative distribution functions and define the following [what I think is a] metric: $d(F,G)$ is the area of the largest rectangle, with sides parallel to the axes, inscribed between the graphs of $F$ and $G$. Has anyone seen such an object?
Update: I'll even conjecture a relationship between $d(F,G)$ and the Levy metric $L(F,G)$: $$ d(F,G) \le L(F,G) \le \sqrt{d(F,G)}. $$
Update II: Following Gil's suggestion, here the definition of Levy metric: https://en.wikipedia.org/wiki/L%C3%A9vy_metric
In words, it's what you get if you replace "rectangle" by "square" and "area" by "side length" in my definition. I should say that the definition of $d(\cdot)$ is due to Eyal Shimony.