On sets defining few ordinary planes

2016/06/03 by Ball, Simeon
#51M04 #52C35 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1606.02138

Abstract

Let S be a set of n points in real three-dimensional space, no three collinear and not all co-planar. We prove that if the number of planes incident with exactly three points of S is less than Kn² for some K=o(n^(1)/(7)) then, for n sufficiently large, all but at most O(K) points of S are contained in the intersection of two quadrics. Furthermore, we prove that there is a constant c such that if the number of planes incident with exactly three points of S is less than (1)/(2)n²-cn then, for n sufficiently large, S is either a prism, an anti-prism, a prism with a point removed or an anti-prism with a point removed. As a corollary to the main result, we deduce the following theorem. Let S be a set of n points in the real plane. If the number of circles incident with exactly three points of S is less than Kn² for some K=o(n^(1)/(7)) then, for n sufficiently large, all but at most O(K) points of S are contained in a curve of degree at most four.

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