Convexity/Helly's theorem

Helly's theorem is as follows:

Let Rⁿ be an n-dimensional real vector space and let N > n. Suppose you have N convex sets in Rⁿ such that, given any n+1 of them, they have at least one point in common. Then all N sets have at least one point in common.

Proof: Obviously, the theorem is true for N = n+1. Proceed by induction and assume it is true for N-1. Let the sets be X₁ to X_N. For each i, exclude set X_i. By the inductive hypothesis, there is at least one point x(i) belonging to all the sets except possibly X_i. Since N > n+1, the (n+1) equations

$\displaystyle \sum _{i=1}^{N}\lambda _{i}x_{k}(i)=0$

$\displaystyle (k=1,2,...n)$

$\displaystyle \sum _{i=1}^{N}\lambda _{i}=0$

in the N unknowns λ₁ to λ_N must have non-trivial solutions. For one such solution, denote by λ_j1 to λ_jk those λ that are positive, and λ_h1 to λ_h(N-k) those that are negative.