On distinct distances among points in general position and other related problems |
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Authors: | Adrian Dumitrescu |
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Institution: | (1) Department of Computer Science, University of Wisconsin-Milwaukee, Milwaukee, WI 53201-0784, USA |
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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 Erd?s, Hickerson and Pach. We then revisit an old problem of Erd?s: 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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Keywords: | and phrases" target="_blank"> and phrases distinct distances distinct vectors general position Sidon sequence |
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