A missing prime configuration in the Hausdorff metric geometry
The paper deals with the geometry of the space \(\mathcal{H}(\mathbb{R}^n)\), which is the hyperspace of non-empty compact subsets of \(\mathbb{R}^n\) endowed with the Hausdorff metric \(h\). The main topic is the study of the betweenness relation. Given two distinct elements \(A\) and \(B\) in \(\mathcal{H}(\mathbb{R}^n)\), an element \(C\) in \(\mathcal{H}(\mathbb{R}^n)\) is said to lie between \(A\) and \(B\) (denoted by \(ACB\)), if \(h(A,B) = h(A,C) + h(C,B)\). If \(A\) and \(B\) are given distinct elements, two elements \(C\) and \(C'\) are at the same location between \(A\) and \(B\) if \(C\) and \(C'\) satisfy \(ACB\) and \(AC'B\) with \(h(A,C) = h(A,C') =t\) for some \(0<t<h(A,B)\). The main question that is considered is which finite numbers can arise as numbers of elements that can exist at each location beween elements \(A\) and \(B\). The authors prove that if there are only finitely many elements at each location between \(A\) and \(B\), then every point in \(A\) is at the same distance from \(B\) and every point in \(B\) is at the same distance from \(A\). This observation leads to an appropriate notion of ``configuration \([A,B]\)''. For infinitely many values of the positive integer \(k\), there is a configuration so that there exist exactly \(k\) elements at each location between \(A\) and \(B.\) It is shown that if \(k\) lies between \(1\) and \(18\), there is always a configuration having exactly \(k\) elements at each location between the end elements, however for \(k=19\) it is shown that no such configuration exists.
scientific article; zbMATH DE number 5562634