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Archiv der Mathematik

, 97:289 | Cite as

On the optimal interpoint distance sum inequality

  • Stefan SteinerbergerEmail author
Article
  • 91 Downloads

Abstract

Let \({{\left\{x_{1}, \dots, x_{n}\right\}\subset \mathbb{R}^2}}\) be a set of points in the unit circle. It is shown that
$$\sum\limits^{n}_{i=1}{\min_{j \neq i}{\left\|x_{i} - x_{j}\right\|^2}}\leq9,$$
which is best possible and improves earlier results by Arpacioglu and Haas and Xia and Liu.

Mathematics Subject Classification (2010)

51M16 52C15 

Keywords

Circle packing 

References

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    Witsenhausen H.: On the maximum of the sum of squared distances under a diameter constraint. Amer. Math. Monthly 81, 1100–1101 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
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    Y. Xia and H. Liu, Improving upper bound on the capacity of planar wireless networks with omnidirectional antennas, in: B. Yuan and X. Tang (eds.) Proceedings of the IET 2nd International Conference on Wireless, Mobile & Multimedia Networks, 191–194, 2008.Google Scholar
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    Y. Xia and H. Liu, On the interpoint distance sum inequality, JIPAM. J. Inequal. Pure Appl. Math. 10, no. 3, Article 74, 10 pp (2009).Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of BonnBonnGermany

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