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Helly's theorem
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(Theorem)
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Suppose
 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 $I=(\conv P_1)\cap(\conv P_2)\neq \emptyset$ . 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 $j$ . 
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"Helly's theorem" is owned by bbukh.
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Cross-references: belongs, partition, Radon's lemma, point, inductive hypothesis, vacuous, induction, proof, intersection, convex sets
There is 1 reference to this entry.
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 5063 times total.
Classification:
| AMS MSC: | 52A35 (Convex and discrete geometry :: General convexity :: Helly-type theorems and geometric transversal theory) |
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Pending Errata and Addenda
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