2008/03/07 by Ramos, Pedro, Viaña, Raquel · 1 citation
#52C35 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.0803.1088
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 R3, 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.