Radon's theorem
/ RAH-dohn /
Drop a few too many points into n-dimensional space and they cannot help getting in each other's way. Radon's theorem makes this precise: any n + 2 points in R^n can always be split into two groups whose convex hulls overlap. No matter how you scatter four points in the plane, you can color them red and blue so that the red region and the blue region share a point. There is an unavoidable 'crossing' once you have two more points than the dimension.
Precisely, given any n + 2 points x_1, ..., x_{n+2} in R^n, there is a partition of the index set into two disjoint parts I and J such that conv{x_i : i in I} and conv{x_j : j in J} have a common point. Such a partition is called a Radon partition, and the shared point a Radon point. The proof is pure linear algebra: n + 2 vectors in R^n, lifted to R^{n+1} by appending a 1, are linearly dependent, so there are coefficients a_i, not all zero, with sum a_i x_i = 0 and sum a_i = 0. Put the indices with a_i > 0 in I and those with a_i < 0 in J; normalizing the positive and negative parts gives the same point as a convex combination from each side — that point lies in both hulls.
Radon's theorem is the engine behind Helly's theorem (the standard proof of Helly inducts using Radon partitions) and a member of the combinatorial-convexity trio with Helly and Caratheodory. It also generalizes spectacularly: Tverberg's theorem says that (r-1)(n+1) + 1 points in R^n can be split into r groups all of whose hulls share a common point, with Radon the case r = 2. These results underlie centerpoint theorems and have surprising applications in combinatorics and the topology of the chessboard complex. An honest caveat: the partition is guaranteed to exist but is not unique, and Radon says nothing about WHERE the common point is or how to find it efficiently in high dimensions beyond the linear-algebra construction; the points must also be allowed to be in general position or not — the theorem holds regardless, but degenerate configurations can make several partitions valid at once.
Four points in the plane (n = 2, so n + 2 = 4) always admit a Radon partition. If the four are the corners of a convex quadrilateral ABCD, split them as the two diagonals {A, C} and {B, D}: the segments AC and BD cross, so the two hulls (the diagonals) share the crossing point. If instead one point D lies inside triangle ABC, split as {A, B, C} and {D}: the triangle contains D, so the hulls again meet — at D itself.
Four planar points split into two hulls that meet: crossing diagonals, or a point inside a triangle.
The Radon partition exists but need not be unique, and the theorem is silent on locating the common point efficiently beyond the linear-dependence construction. It is an existence result; Tverberg's theorem is the deep multi-part generalization, far harder to prove.