There are two sets that will play
a vital role in understanding the geometry the Hausdorff metric imposes on the
space *H*(**R*** ^{n}*).
The first is the dilation of a set and the second is the neighborhood of a set.

**Definition: ***Let A be an element of H*(**R*** ^{n}*)

(*A*)* _{r}
*= {

Informally, we can obtain the dilation
of the set *A* by *r *by stretching *A *by *r* units in all
directions. We will see that dilations are important in creating Hausdorff
circles in the next section.

**Definition: ***Let A be an element of H*(**R*** ^{n}*)

*N _{r}*(

Note that *N _{r}*(

Figure 1: (*A*)* _{r}* (left) and