A Missing Prime Configuration in the Hausdorff Metric Geometry

Abstract.

The Hausdoff metric h was introduced by Felix Hausdoff in the early 20th century as a way to measure the distance between elements in the hyperspace ${\mathcal{H}}({\mathbb{R}}^{n})$ of non-empty compact subsets of ${\mathbb{R}}^{n}$. The geometry this metric imposes on ${\mathcal{H}}({\mathbb{R}}^{n})$ has many interesting properties: lines in this geometry can have endpoints; there can be many elements at a given location between two sets in ${\mathcal{H}}({\mathbb{R}}^{n})$; the Fibonacci and Lucas numbers arise in a natural way in this geometry; and for infinitely many different values of the positive integer k, there is a configuration [A, B] in ${\mathbb{R}}^{n}$ so that there exist exactly k elements at each location between A and B. In this paper we will show that if k is between 1 and 18, there is always a configuration having exactly k elements at each location between the end elements and prove the surprising result that no such configuration exists if k equals 19.