The Strange World of the Hausdorff Metric Geometry
XXI. Summary
The goal of this paper was to
give a short introduction to the strange world of the Hausdorff metric. Some of
the odd behavior that we have seen includes
·
Given A and B in H(Rn), there may be more than one element C in
H(Rn) at a
given location satisfying ACB, ABC, or CAB.
·
Given any positive integer k, there exist A, B in
H(Rn) so
that the number of elements C in H(Rn) satisfying ACB is the kth
Fibonacci number, the kth Lucas number, any k between
1 and 18, any k between 20 and 36,
but there are no A, B in H(Rn) so that the number of elements C in H(Rn) satisfying ACB
is exactly 19 or 37.
·
There are some integers that appear as #([A, B]) in H(R2) but not in H(R),
some that appear in H(R3) but not H(R2).
·
For some sets A and B,
there can be finite sets at some locations between A and B and
infinite sets at other locations.
·
It is possible for #([A, B]) to be finite and
#([A, B]) to be uncountably infinite, but it is impossible for #([A, B])
to be countably infinite.
·
Hausdorff lines do not always contain elements at each location.
They can actually stop. When a Hausdorff line stops, it is really a halfline.
As you can see, the Hausdorff
metric geometry is often counter-intuitive and usually fascinating.
Questions for
Future Research
Among the many open questions in
the Hausdorff metric geometry are the following.
·
The standard Euclidean geometry is embedded in the geometry of H(Rn) as the single
element sets. However, the geometry of H(Rn) is not itself an incidence geometry. Are
there subspaces of H(Rn),
other than the standard Euclidean geometry, or are there certain types of lines
for which the resulting geometry is an incidence geometry?
·
What other subspaces of H(Rn) have interesting structures? For example,
what, if anything, can we say about the geometry of H(Tn),
where Tn is the n-torus?
·
For which positive integers k can we find A and B
in H(Rn)
so that the number of elements C in H(Rn) at a given location satisfying ACB is
exactly k?
·
For any sets A and B in H(Rn), can we completely classify the elements C
in H(Rn)
that satisfy ACB, ABC, and/or CAB?
·
Can we completely determine the partition of the positive integers
into SPACK classes?