nLab
Hausdorff metric

The Hausdorff metric

Idea

The Hausdorff metric is a metric on the power set of a given metric space.

Definition

Let AA be a metric space, regarded as a category enriched over V=[0,]V = [0,\infty] (a Lawvere metric space). The enriched functor category [A,V][A,V] is, concretely, the set of short maps A[0,]A \to [0,\infty] with the supremum metric?, and the contravariant Yoneda embedding A op[A,V]A^{op} \to [A,V] sends aa to d(a,)d(a,-).

Now, for any subset XAX \subseteq A, each point xXx \in X gives rise to the representable functor d(x,)d(x,-), and we can define the functor d(X,):AVd(X,-)\colon A \to V to be the coproduct of these representables over all xXx \in X. Concretely, this means

d(X,a)=inf xXd(x,a). d(X,a)=inf_{x\in X} d(x,a) .

Note that if XX is closed, then it can be recovered from d(X,)d(X,-) as the set of points xx such that d(X,x)=0d(X,x)=0. If XX is not closed, then in this way we recover its closure.

Finally, since [A,V][A,V] is also a Lawvere metric space, we obtain an induced metric on the set of subspaces of AA:

d(X,Y)=d(d(X,),d(Y,)). d(X,Y) = d(d(X,-), d(Y,-)) .

This metric is not symmetric (so a quasimetric); its symmetrization? is the Hausdorff metric. Equivalently, we could start out by considering instead the functor category [A,V sym][A, V_{sym}] where V symV_{sym} is the symmetrization of V=[0,]V = [0,\infty].

References

  • Taking categories seriously, Reprints in Theory and Applications of Categories, No. 8, 2005, pp. 1–24. (pdf)
Revised on September 7, 2011 04:04:47 by Toby Bartels (75.88.82.16)