So I have a simple question regarding convex sets and their intersection.
Let $\{ X_i \}_{i \in I}$ be a collection of open convex sets in $\mathbb{R}^m$ such that $X_i \cap X_j \cap X_k \neq \emptyset$ for all $i,j,k \in I$.
Does it then necessarily follow that $\bigcap_{i \in I} X_i \neq \emptyset$?
If it does, what is the proof, and if it does not, what is a good counterexample?
I look forward to your answers.
For finite $I$ it’s true in $\Bbb R$ and $\Bbb R^2$, thanks to Helly’s theorem, but without compactness it can fail for infinite $I$ even in $\Bbb R$, as user587399 pointed out in the comments. (We can, for instance, let $I_n=(0,2^{-n})$.) It also fails in $\Bbb R^m$ for $m\ge 3$. In $\Bbb R^3$, for instance, start with four points on the unit sphere located at the vertices of a regular tetrahedron. The edge length of the tetrahedron is $\sqrt{8/3}$, so the distance from a vertex to the centre of any face of which it is a vertex is $\frac{2\sqrt2}3$, so choose $r\in\left(\frac{2\sqrt2}3,1\right)$; then any three of the open balls of radius $r$ centred at the vertices of the tetrahedron intersect, but the intersection of all four is empty.