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 extension 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 extension of A by r is the set

A + r = {x in Rⁿ : d_E(a, x) ≤ r for some a in A}.

Informally, we can obtain the extension of the set A by r by stretching A by r units in all directions. We will see that extensions 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

N_r(A) = {x in Rⁿ : d_E(a, x) < r for some a in A}.

Note that N_r(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 N_r(A) when A is the union of two Euclidean line segments.

Figure 1: A+r (left) and N_r(A) (right)