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On the optimal interpoint distance sum inequality

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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.

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References

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Authors and Affiliations

  1. Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Stefan Steinerberger

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  1. Stefan Steinerberger
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Correspondence to Stefan Steinerberger.

Additional information

This work was supported by the Austrian Science Foundation (FWF), Project S9609, which is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.

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Steinerberger, S. On the optimal interpoint distance sum inequality. Arch. Math. 97, 289–298 (2011). https://doi.org/10.1007/s00013-011-0293-7

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  • DOI: https://doi.org/10.1007/s00013-011-0293-7

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