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*(**R*** ^{n}*), there may be more than one element

·
Given any positive integer *k*, there exist *A*, *B *in
*H*(**R*** ^{n}*) so
that the number of elements

·
There are some integers that appear as #([*A*, *B*]) in *H*(**R**^{2}) but not in *H*(**R**),
some that appear in *H*(**R**^{3}) but not *H*(**R**^{2}).

·
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.

Among the many open questions in
the Hausdorff metric geometry are the following.

·
The standard Euclidean geometry is embedded in the geometry of *H*(**R*** ^{n}*) as the single
element sets. However, the geometry of

·
What other subspaces of *H*(**R*** ^{n}*) have interesting structures? For example,
what, if anything, can we say about the geometry of

·
For which positive integers *k *can we find *A* and *B
*in *H*(**R*** ^{n}*)
so that the number of elements

·
For any sets *A* and *B* in *H*(**R*** ^{n}*), can we completely classify the elements

·
Can we completely determine the partition of the positive integers
into SPACK classes?