On the sets of $n$ points forming $n+1$ directions

Abstract

Let $S$ be a set of $n\geq 7$ points in the plane, no three of which are collinear. Suppose that $S$ determines $n+1$ directions. That is to say, the segments whose endpoints are in $S$ form $n+1$ distinct slopes. We prove that $S$ is, up to an affine transformation, equal to $n$ of the vertices of a regular $(n+1)$-gon. This result was conjectured in 1986 by R. E. Jamison. In an addendum to the paper, we show that a much stronger result can be obtained as a corollary of a structure theorem of Green and Tao on point sets spanning few ordinary lines.