# Convexity/Helly's theorem

**Helly's theorem** is as follows:

Let R^{n} be an n-dimensional real vector space and let N > n. Suppose you have N convex sets in R^{n} 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 X_{1} 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

in the N unknowns λ_{1} 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.

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