The Strange World of the Hausdorff Metric Geometry

VII. Simple Examples: One-Point Sets

To illustrate the geometry, we begin with the simplest elements in H(Rⁿ), namely the one point sets. In the 2002 REU, John Mayberry (then at the University of California at Fullerton, now at the University of Southern California), Audrey Powers (then at Agnes Scott College, now at Emory University) and I examined the subspace of H(Rⁿ) consisting of the one-point sets. We completely classified the elements in H(R ⁿ) that satisfy the three versions of the triangle equality in this subspace. The next theorem (Theorem 6 from [7]) provides the classification.

Theorem 2: Let a, b be points in Rⁿ. Let A = {a}, B = {b} and C be elements of H(Rⁿ). Then C satisfies ABC, ACB, or CAB

if and only if there exist positive real numbers s and t so that

1. C is a subset of (A+s) ∩ (B+t) and

2. there is a point c₀ in C with c₀ in the intersection of ∂(A+s) ∩ ∂(B+t) with the Euclidean line through a and b.

This theorem is illustrated by the next applet. Any compact set in the shaded region that contains the black point c₀ on the ab axis is an element on the Hausdorff line defined by A and B. The points a and b are directly moveable as is the point c₀, by moving the black point on the slider.

One interesting consequence of Theorem 2 is that there many be many elements at a particular location on a Hausdorff line. For example, if the point c₀ described in Theorem 2 is on the Euclidean segment defined by a and b, note that the intersection of A+s with B+t is the single point set C = {c₀}. In this case, there is exactly one point at this location on the Hausdorff line defined by A and B. However, if c₀ lies outside the Euclidean interval defined by a and b, then the intersection of A+s with B+t is a disk with positive radius. In this case, there are many different compact sets contained in the intersection of A+s with B+t. The union of any such set with {c₀} lies s units from A and t units from B. We will say that these sets lie on the same location Hausdorff line defined by A and B.

Definition: Let A and B be distinct elements of H(Rⁿ) and let r = h(A, B). Two elements C₁ and C₂ of H(Rⁿ) on the Hausdorff line L(A, B), through A and B lie at the same location on L(A, B) if

h(A, C₁) = h(A, C₂) and h(B, C₁) = h(B, C₂).

Unlike Euclidean lines which have unique points at each location, Theorem 2 tells us that Hausdorff lines may contain an infinite number or a finite number of elements at a given location. We will see more about this later.