The Average Number of Points in a Spanned Hyperplane

A classic theorem in discrete geometry, the Sylvester-Gallai Theorem, says that for any finite, non-collinear set of points in the plane, there is a (spanned) line containing exactly two of them. A stronger result, due to Melchior, says that a spanned line intersects such a point-set in fewer than three points on average. In this talk, I will sketch the proof of the following three-dimensional version of Melchior’s theorem: given a finite set of points in 3-space, the average number of points in a spanned plane is at most an absolute constant, unless that point-set is coplanar or contained in the union of two lines (in which cases the average could be arbitrarily large). I will also present a generalization of this result to hyperplanes of arbitrary dimension, and discuss its connection to an old problem of Motzkin, Grünbaum, Erd\H{o}s and Purdy concerning red-blue arrangements of points in the plane.

This talk is based on joint works with Rutger Campbell, Jim Geelen and Ben Lund.