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Concept
Helly's theorem
Summary
Helly's theorem is a basic result in discrete geometry on the intersection of convex sets. It was discovered by Eduard Helly in 1913, but not published by him until 1923, by which time alternative proofs by and had already appeared. Helly's theorem gave rise to the notion of a Helly family.
Let X1, ..., Xn be a finite collection of convex subsets of Rd, with n ≥ d + 1. If the intersection of every d + 1 of these sets is nonempty, then the whole collection has a nonempty intersection; that is,
For infinite collections one has to assume compactness:
Let {Xα} be a collection of compact convex subsets of Rd, such that every subcollection of cardinality at most d + 1 has nonempty intersection. Then the whole collection has nonempty intersection.
We prove the finite version, using Radon's theorem as in the proof by . The infinite version then follows by the finite intersection property characterization of compactness: a collection of closed subsets of a compact space has a non-empty intersection if and only if every finite subcollection has a non-empty intersection (once you fix a single set, the intersection of all others with it are closed subsets of a fixed compact space).
The proof is by induction:
Base case: Let n = d + 2. By our assumptions, for every j = 1, ..., n there is a point xj that is in the common intersection of all Xi with the possible exception of Xj. Now we apply Radon's theorem to the set A = {x1, ..., xn}, which furnishes us with disjoint subsets A1, A2 of A such that the convex hull of A1 intersects the convex hull of A2. Suppose that p is a point in the intersection of these two convex hulls. We claim that
Indeed, consider any j ∈ {1, ..., n}. We shall prove that p ∈ Xj. Note that the only element of A that may not be in Xj is xj. If xj ∈ A1, then xj ∉ A2, and therefore Xj ⊃ A2. Since Xj is convex, it then also contains the convex hull of A2 and therefore also p ∈ Xj. Likewise, if xj ∉ A1, then Xj ⊃ A1, and by the same reasoning p ∈ Xj. Since p is in every Xj, it must also be in the intersection.
Official source
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