Hausdorff distance

The Hausdorff metric, named after the mathematician Felix Hausdorff measures, the distance δ (A, B) between non-empty compact subsets A, B of a metric space E.

Clearly have two compact subsets more lower Hausdorff distance the better they cover, are interdependent.

Definition

As an aid to define the distance between a point and a non-empty compact subset with recourse to the metric of the space as

Then we define the Hausdorff distance between two nonempty compact subsets and as

One can show that in fact is a metric on the set of all compact subsets of.

Applications

In the theory of iterated function systems fractals are generated as a result of limits in terms of the Hausdorff metric.