A theorem giving a useful property of convexity in Rd. 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 Rd. 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
R3 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.