On the number of ordinary circles

2014/12/29 by Mojarrad, Hossein Nassajian, de Zeeuw, Frank
#Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1412.8314

Abstract

We prove that any n points in ℝ², not all on a line or circle, determine at least (1)/(4)n²-O(n) ordinary circles (circles containing exactly three of the n points). The main term of this bound is best possible for even n. Our proof relies on a recent result of Green and Tao on ordinary lines.

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