Hausdorff metric

Let $(X,d)$ be a metric space, and let ${ℱ}_X$ be the family of allclosed and bounded subsets of $X$. Given $A\in {ℱ}_X$, 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 $A,B\in {ℱ}_X$.

The following properties follow straight from the definitions:

1. ${\delta }^{*}(A,B)={\delta }_{*}(B,A)$;

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

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

4. ${\delta }^{*}(A,C)\le {\delta }^{*}(A,B)+{\delta }^{*}(B,C)$, 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 $({ℱ}_X,\delta )$. Also, if $(X,d)$ is totally bounded, then so is$({ℱ}_X,\delta )$.

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.