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arXiv · 0803.1088

Depth of segments and circles through points enclosing many points: a note

Pedro Ramos·Raquel Viaña
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Abstract

Neumann-Lara and Urrutia showed in 1985 that in any set of n points in the plane in general positionthere is always a pair of points such that any circle through them contains at least (n-2)/60 points. In a series of papers, this result was subsequently improved till n/4.7, which is currently the best known lower bound. In this paper we propose a new approach to the problem that allows us, by using known results about j-facets of sets of points in $R^3$, to give a simple proof of a somehow stronger result: there is always a pair of points such that any circle through them has, both inside and outside, at least n/4.7 points.

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Pedro Ramos, Raquel Viaña. 2008-03-07. Depth of segments and circles through points enclosing many points: a note. https://arxiv.org/abs/0803.1088

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