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.