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On distinct distances among points in general position and other related problems

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Abstract

A set of points in the plane is said to be in general position if no three of them are collinear and no four of them are cocircular. If a point set determines only distinct vectors, it is called parallelogram free. We show that there exist n-element point sets in the plane in general position, and parallelogram free, that determine only O(n 2/√log n) distinct distances. This answers a question of Erdős, Hickerson and Pach. We then revisit an old problem of Erdős: given any n points in the plane (or in d dimensions), how many of them can one select so that the distances which are determined are all distinct? — and provide (make explicit) some new bounds in one and two dimensions. Other related distance problems are also discussed.

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  1. Department of Computer Science, University of Wisconsin-Milwaukee, Milwaukee, WI, 53201-0784, USA

    Adrian Dumitrescu

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Correspondence to Adrian Dumitrescu.

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Supported in part by NSF CAREER grant CCF-0444188.

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Dumitrescu, A. On distinct distances among points in general position and other related problems. Period Math Hung 57, 165–176 (2008). https://doi.org/10.1007/s10998-008-8165-4

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  • DOI: https://doi.org/10.1007/s10998-008-8165-4

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