Let $\displaystyle X_1,\ldots ,X_n\subseteq \mathbb{R}^m$ be convex open sets such that $\displaystyle X_i\cap X_j\cap X_k\neq \emptyset$ for every $\displaystyle i,j,k$. Show that $\displaystyle \bigcap_{i=1}^n X_i\neq \emptyset$.
Let $\displaystyle X_1,\ldots ,X_n\subseteq \mathbb{R}^m$ be convex open sets such that $\displaystyle X_i\cap X_j\cap X_k\neq \emptyset$ for every $\displaystyle i,j,k$. Show that $\displaystyle \bigcap_{i=1}^n X_i\neq \emptyset$.
you can find a proof on Helly's theorem - Wikipedia, the free encyclopedia but it is a special case. i don't know if the statement i wrote is true. The question is open.