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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?

 

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