Suppose is a family of convex sets, and every of them have a non-empty intersection. Then is non-empty.
The proof is by induction on . If , then the statement is vacuous. Suppose the statement is true if is replaced by . The sets are non-empty by inductive hypothesis. Pick a point from each of . By Radon’s lemma, there is a partition of ’s into two sets and such that . For every either every point in belongs to or every point in belongs to . Hence for every . ∎