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The Strange World of the Hausdorff Metric Geometry


XVIII. Other Properties of Betweenness


We have discussed many properties of betweenness in H(Rn) and we conclude with three more.


For infinitely many different values of k and for all k between 1 and 18 and 20 through 36,  there are configurations [A, B] with #([A, B]) = k. In addition, if A and B do not form a configuration, we know that the number of sets between A and B at each location is uncountably infinite. An interesting question is whether there can be a configuration [A, B] so that #([A, B])  is countably infinite. In [26], Alex Zupan (REU 2005) shows that the answer to this question is no.


If  [A, B] is a finite configuration, then the sets between A and B at each location are finite, while if A and B do not form a configuration, then we have infinite sets that lie between A and B. Another interesting question, answered by David Montague (REU 2008) in [16] is whether there can be finite sets at some locations between A and B and infinite sets at other locations. The answer, surprisingly, turns out to be yes.


Finally, given any two compact sets A and B, we know the set M(s) = (A)s (B)h(A,B)-s lies between A and B at the distance s from A. Therefore, the space in H(Rn) is a convex space. Vincent Martinez (REU 2007) considered different types of convexity in H(Rn) and arrived at some interesting results worthy of further investigation.


I hope to provide more details about these particular topics (and others) in the future.


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