A theorem giving a useful property of convexity in **R**^{d}. Besides being related to Caratheodory's theorem and Helly's theorem, it has interesting analogues in the algebraic characterisation of dimension as the maximal number of linearly independent points, or as 1 less than the maximal number of affinely independent points.

**Theorem.** Let V be a set of n>d+1 points in **R**^{d}. Then V may be partitioned into two disjoint sets whose convex hulls intersect.

See a proof of Radon's theorem, if you like, although it's really just an exercise in affine dependence.

In

**R**^{3} any 4 points

*not* on one

plane have no such partition. But given 5 points, the theorem says either one of them is inside the

tetrahedron formed by the other 4, or 3 of them form a

triangle and the

segment connecting the other 2 passes through that triangle.