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

14 years 28 days ago

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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(n2 / log n) distinct distances. This answers a question of Erdos, Hickerson and Pach. We then revisit an old problem of Erdos : 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.

Adrian Dumitrescu

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