It is also an extremely useful mathematical concept. To define it in a general metric space, you need to replace the min with inf, of course. Intuitively, Hausdorff distance is the furthest away you can be in one of A,B from the other.

As an example of what can be done with this distance measure, the set of convex polygons is dense in the set of convex compact shapes in the plane. And the set of all convex compact shapes is a complete metric space when equipped with this metric! Area and perimeter of convex compact shapes are continuous functions with this metric, too.