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 , 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  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.