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
$$\delta (A,B)=\mathrm{max}\{{\delta}^{*}(A,B),{\delta}_{*}(A,B)\}.$$ 
for $A,B\in {\mathcal{F}}_{X}$.
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 $({\mathcal{F}}_{X},\delta )$. 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 

Canonical name  HausdorffMetric 
Date of creation  20130322 13:28:34 
Last modified on  20130322 13:28:34 
Owner  Koro (127) 
Last modified by  Koro (127) 
Numerical id  11 
Author  Koro (127) 
Entry type  Definition 
Classification  msc 54E35 
Synonym  Hausdorff distance 
Defines  Hausdorff hemimetric 