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Helly's theorem (Theorem)

Suppose $ A_1,\dotsc,A_m\subset \mathbb{R}^d$ is a family of convex sets, and every $ d+1$ of them have a non-empty intersection. Then $ \bigcap_{i=1}^m A_i$ is non-empty.

Proof. The proof is by induction on $ m$. If $ m=d+1$, then the statement is vacuous. Suppose the statement is true if $ m$ is replaced by $ m-1$. The sets $ B_j=\bigcap_{i\neq j} A_i$ are non-empty by inductive hypothesis. Pick a point $ p_j$ from each of $ B_j$. By Radon's lemma, there is a partition of $ p$'s into two sets $ P_1$ and $ P_2$ such that $ I=({\mathrm{conv}}P_1)\cap({\mathrm{conv}}P_2)\neq \emptyset$. For every $ A_j$ either every point in $ P_1$ belongs to $ A_j$ or every point in $ P_2$ belongs to $ A_j$. Hence $ I\subseteq A_j$ for every $ j$. $ \qedsymbol$



"Helly's theorem" is owned by bbukh.
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Cross-references: partition, Radon's lemma, point, inductive hypothesis, vacuous, induction, proof, intersection, convex sets
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This is version 3 of Helly's theorem, born on 2003-09-12, modified 2003-11-16.
Object id is 4728, canonical name is HellysTheorem.
Accessed 4148 times total.

Classification:
AMS MSC: 52A35 (Convex and discrete geometry :: General convexity :: Helly-type theorems and geometric transversal theory)
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