There are two sets that will play
a vital role in understanding the geometry the Hausdorff metric imposes on the
space H(Rn).
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(Rn) and let r > 0. The dilation of A by r is the set
(A)r
= {x in Rn : dE(a, x) ≤ r for some a in A}.
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(Rn) and let r > 0. The r-neighborhood of A is the set
Nr(A) = {x in Rn
: dE(a, x) < r for some a in A}.
Note that Nr(A)
is the union of all the open r-neighborhoods of the points in A
and, as a result, is an open set. The extension A+r is, by
contrast, a compact set. Figure 1 shows (A)r and Nr(A) when A is the union of two Euclidean line
segments.
Figure 1: (A)r (left) and Nr(A)
(right)