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, Audrey Powers, 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 may 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 on the 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.