Helly's Theorem

If $F$ is a family of more than $n$ bounded closed convex sets in Euclidean $n$-space $\Bbb{R}^n$, and if every $H_n$ (where $H_n$ is the Helly Number) members of $F$ have at least one point in common, then all the members of $F$ have at least one point in common.

See also Carathéodory's Fundamental Theorem, Helly Number