Infinite cardinalities in the Hausdorff metric geometry (Q399446)

Let \({\mathcal H}({\mathbb R}^n)\) be the collection of nonempty compact sets in \({\mathbb R}^n\) and \(h\) be the Hausdorff metric on \({\mathcal H}({\mathbb R}^n)\). For any \(A, B, C \in {\mathcal H}({\mathbb R}^n)\) and \(0<t < h(A,B)\), we say that \(C\) is between \(A\) and \(B\) at a distance \(t\) from \(A\) if NEWLINENEWLINE\[NEWLINEh(A, B) = h(A, C) + h (C, B)\;\text{ and }\;h(A, C) =t.NEWLINE\]NEWLINE NEWLINENote, that this concept of betweenness in \({\mathcal H}({\mathbb R}^n)\) is a natural extension of that in \({\mathbb R}^n\). However, in \({\mathbb R}^n\), there is a unique point between \(a\) and \(b\) at a given distance less than \(d(a,b)\) from \(a\), while in \({\mathcal H}({\mathbb R}^n)\) there can be many distinct elements between elements \(A\) and \(B\) at a given distance from \(A\). NEWLINE\textit{C. C. Blackburn} et al. [J. Geom. 92, No. 1--2, 28--59 (2009; Zbl 1171.54008)] demonstrated that there are infinitely many positive integers \(k\) such that there exist elements \(A\) and \(B\) in \({\mathcal H}({\mathbb R}^n)\) having exactly \(k\) different elements between \(A\) and \(B\) at each distance from \(A\) while proving a surprising result that no such \(A\) and \(B\) exist for \(k=19\). NEWLINEIn this paper, the author proves that there do not exist elements \(A\) and \(B\) in \({\mathcal H}({\mathbb R}^n)\) with exactly a countably infinite number of elements at any location between \(A\) and \(B\).