A Product Inequality for Extreme Distances

Author Adrian Dumitrescu

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Author Details

Adrian Dumitrescu
  • Department of Computer Science, University of Wisconsin - Milwaukee, USA

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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)


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
  • Extreme distances
  • repeated distances


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  1. Paul Erdös and János Pach. Variation on the theme of repeated distances. Combinatorica, 10(3):261-269, 1990. URL: http://dx.doi.org/10.1007/BF02122780.
  2. Heiko Harborth. Solution to problem 664A. Elem. Math, 29:14-15, 1974. Google Scholar
  3. János Pach and Pankaj K. Agarwal. Combinatorial Geometry. Wiley-Interscience series in discrete mathematics and optimization. Wiley, 1995. Google Scholar
  4. Paul R. Scott and Poh Wah Awyong. Inequalities for convex sets. Journal of Inequalities in Pure and Applied Mathematics, 1(1):6, 2000. Google Scholar
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