Hausdorff metric

Let $(X,d)$ be a metric space, and let $\mathcal{F}_{X}$ be the family of all closed and bounded subsets of $X$. Given $A\in\mathcal{F}_{X}$, we will denote by $N_{r}(A)$ the neighborhood of $A$ of radius $r$, i.e. the set $\cup_{x\in A}B(x,r)$.

The upper Hausdorff hemimetric is defined by

 $\delta^{*}(A,B)=\inf\{r>0:B\subset N_{r}(A)\}.$

Analogously, the lower Hausdorff hemimetric is

 $\delta_{*}(A,B)=\inf\{r>0:A\subset N_{r}(B)\}.$

Finally, the Hausdorff metric is given by

 $\delta(A,B)=\max\{\delta^{*}(A,B),\delta_{*}(A,B)\}.$

for $A,B\in\mathcal{F}_{X}$.

The following properties follow straight from the definitions:

1. 1.

$\delta^{*}(A,B)=\delta_{*}(B,A)$;

2. 2.

$\delta^{*}(A,B)=0$ if and only if $B\subset A$;

3. 3.

$\delta_{*}(A,B)=0$ if and only if $A\subset B$;

4. 4.

$\delta^{*}(A,C)\leq\delta^{*}(A,B)+\delta^{*}(B,C)$, and similarly for $\delta_{*}$.

From this it is clear that $\delta$ is a metric: the triangle inequality follows from that of $\delta_{*}$ and $\delta^{*}$; symmetry follows from $\delta^{*}(A,B)=\delta_{*}(A,B)$; and $\delta(A,B)=0$ iff both $\delta_{*}(A,B)$ and $\delta^{*}(A,B)$ are zero iff $A\subset B$ and $B\subset A$ iff $A=B$.

Hausdorff metric inherits completeness; i.e. if $(X,d)$ is complete, then so is $(\mathcal{F}_{X},\delta)$. Also, if $(X,d)$ is totally bounded, then so is $(\mathcal{F}_{X},\delta)$.

Intuitively, the Hausdorff hemimetric $\delta^{*}$ (resp. $\delta_{*}$) measure how much bigger (resp. smaller) is a set compared to another. This allows us to define hemicontinuity of correspondences.

Title Hausdorff metric HausdorffMetric 2013-03-22 13:28:34 2013-03-22 13:28:34 Koro (127) Koro (127) 11 Koro (127) Definition msc 54E35 Hausdorff distance Hausdorff hemimetric