Helly's Theorem

Theorem

Let $A_1, A_2, \ldots, A_r \in \R^n$ be convex sets in real Euclidean $n$-space such that $r > n$.

Let $A_1, A_2, \ldots, A_r$ have the property that every collection of $n + 1$ of $A_1, A_2, \ldots, A_r$ have a point in common.

Then all of $A_1, A_2, \ldots, A_r$ have a point in common.

Proof

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Source of Name

This entry was named for Eduard Helly.

Historical Note

Helly's Theorem was published by Eduard Helly in $1923$.

Sources

• 1923: Ed. Helly: Über Mengen konvexer Körper mit gemeinschaftlichen Punkten (J. Deutsche Math.-Ver Vol. 32: pp. 175 – 176)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Helly's theorem