LIPIcs.SoCG.2019.30.pdf
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A Product Inequality for Extreme Distances
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Adrian Dumitrescu 
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35th International Symposium on Computational Geometry (SoCG 2019)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Computational Geometry (SoCG) - License:
Creative Commons Attribution 3.0 Unported license
- Publication Date: 2019-06-11
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Adrian Dumitrescu. A Product Inequality for Extreme Distances. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 30:1-30:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.SoCG.2019.30
https://doi.org/10.4230/LIPIcs.SoCG.2019.30
Abstract
Let p_1,...,p_n be n distinct points in the plane, and assume that the minimum inter-point distance occurs s_{min} times, while the maximum inter-point distance occurs s_{max} times. It is shown that s_{min} s_{max} <= (9/8)n^2 + O(n); this settles a conjecture of Erdős and Pach (1990).
Subject Classification
ACM Subject Classification
- Theory of computation → Randomness, geometry and discrete structures
Keywords
- Extreme distances
- repeated distances
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References
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Heiko Harborth. Solution to problem 664A. Elem. Math, 29:14-15, 1974.
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János Pach and Pankaj K. Agarwal. Combinatorial Geometry. Wiley-Interscience series in discrete mathematics and optimization. Wiley, 1995.
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Paul R. Scott and Poh Wah Awyong. Inequalities for convex sets. Journal of Inequalities in Pure and Applied Mathematics, 1(1):6, 2000.