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2014 Infinite cardinalities in the Hausdorff metric geometry
Alexander Zupan
Involve 7(5): 585-593 (2014). DOI: 10.2140/involve.2014.7.585

Abstract

The Hausdorff metric measures the distance between nonempty compact sets in n, the collection of which is denoted (n). Betweenness in (n) can be defined in the same manner as betweenness in Euclidean geometry. But unlike betweenness in n, for some elements A and B of (n) there can be many elements between A and B at a fixed distance from A. Blackburn et al. (“A missing prime configuration in the Hausdorff metric geometry”, J. Geom., 92:1–2 (2009), pp. 28–59) demonstrate that there are infinitely many positive integers k such that there exist elements A and B having exactly k different elements between A and B at each distance from A while proving the surprising result that no such A and B exist for k=19. In this vein, we prove that there do not exist elements A and B with exactly a countably infinite number of elements at any location between A and B.

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Alexander Zupan. "Infinite cardinalities in the Hausdorff metric geometry." Involve 7 (5) 585 - 593, 2014. https://doi.org/10.2140/involve.2014.7.585

Information

Received: 22 September 2010; Revised: 23 April 2014; Accepted: 11 May 2014; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1336.51007
MathSciNet: MR3245836
Digital Object Identifier: 10.2140/involve.2014.7.585

Subjects:
Primary: 51F99
Secondary: 54B20

Keywords: betweenness , Hausdorff metric , metric geometry

Rights: Copyright © 2014 Mathematical Sciences Publishers

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