The Strange World of the Hausdorff Metric Geometry

 

IV. Two Important Sets

 

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)