Hausdorff Distance

A metric on the nonempty compact subsets of a metric space defined by the smallest epsilon such that each set is contained in the epsilon-neighborhood of the other; equivalently the maximum of the two directed set-to-set distances.

A metric on the family of nonempty bounded subsets of a metric space defined as the maximum of the two directed suprema of distances from points of one set to the other (equivalently the infimal r such that each set is contained in the r-neighbourhood of the other).