A Product Inequality for Extreme Distances

Dumitrescu, Adrian

doi:10.4230/LIPIcs.SoCG.2019.30

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A Product Inequality for Extreme Distances

Author Adrian Dumitrescu

Part of: Volume: 35th International Symposium on Computational Geometry (SoCG 2019)
Part of: Series: Leibniz International Proceedings in Informatics (LIPIcs)
Part of: Conference: Symposium on Computational Geometry (SoCG)
License: Creative Commons Attribution 3.0 Unported license
Publication Date: 2019-06-11

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LIPIcs.SoCG.2019.30.pdf

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Document Identifiers

DOI: 10.4230/LIPIcs.SoCG.2019.30
URN: urn:nbn:de:0030-drops-104343

Subject Classification

ACM Subject Classification

Theory of computation → Randomness, geometry and discrete structures

Keywords

Extreme distances
repeated distances

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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).

Cite As Get BibTex

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

Author Details

Adrian Dumitrescu

Department of Computer Science, University of Wisconsin - Milwaukee, USA

References

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.
Heiko Harborth. Solution to problem 664A. Elem. Math, 29:14-15, 1974.
János Pach and Pankaj K. Agarwal. Combinatorial Geometry. Wiley-Interscience series in discrete mathematics and optimization. Wiley, 1995.
Paul R. Scott and Poh Wah Awyong. Inequalities for convex sets. Journal of Inequalities in Pure and Applied Mathematics, 1(1):6, 2000.

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