# The Hausdorff metric

## Idea

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

## Definition

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

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

$d\left(X,a\right)={\mathrm{inf}}_{x\in X}d\left(x,a\right).$d(X,a)=inf_{x\in X} d(x,a) .

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

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

$d\left(X,Y\right)=d\left(d\left(X,-\right),d\left(Y,-\right)\right).$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 $\left[A,{V}_{\mathrm{sym}}\right]$ where ${V}_{\mathrm{sym}}$ is the symmetrization of $V=\left[0,\infty \right]$.

## 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)