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Journal of Geometry

, Volume 92, Issue 1–2, pp 28–59 | Cite as

A Missing Prime Configuration in the Hausdorff Metric Geometry

  • Chantel C. Blackburn
  • Kristina Lund
  • Steven Schlicker
  • Patrick Sigmon
  • Alexander Zupan
Article

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.

Mathematics Subject Classification (2000).

51F99 54B20 

Keywords.

Hausdoff metric configuration betweenness nineteen metric geometry 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Chantel C. Blackburn
    • 1
  • Kristina Lund
    • 2
  • Steven Schlicker
    • 3
  • Patrick Sigmon
    • 4
  • Alexander Zupan
    • 5
  1. 1.Department of MathematicsUniversity of ArizonaTucsonArizona
  2. 2.5541 Rivertown Circle SWWyomingUSA
  3. 3.Department of MathematicsGrand Valley State UniversityAllendaleUSA
  4. 4.11641 Broadfield CourtRaleighUSA
  5. 5.Department of MathematicsUniversity of IowaIowa CityUSA

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