Consider Helly Theorem, taken from notes by Igor Pak:
Let $X_1, \dots, X_n \in {\mathbb{R}}^2$ be convex regions in the plane such that any triple interesects $X_i \cap X_j \cap X_k \neq 0$. Then there is a point in all the sets, $X_1 \cap \dots \cap X_n \neq \varnothing$.
This result is not obvious (although Pak's proof is short). However, any explicit collection of sets I build such that three of them intersect, have a clear total intersection. How about this simpler result, also from Pak's book:
Let $P_1, \dots, P_n \in {\mathbb R}^2$ be rectangles with sides parallel to the coordinate axes, such that any two intersect each other. Then all the rectangles have a nonempty intersection.
By Helly's theorem, we only need $n = 3$. What happens if we don't use Helly's theorem and try to prove this result directly?
Let $[x_1, x_1']\times [y_1, y_1'], \dots, [x_n, x_n']\times [y_n, y_n'] \in {\mathbb R}^2$ be rectangles in the plane, sides parallel to the $x,y$-axes, such that:
$x_i < x_j < x_i' < x_j'$ (or vice-versa) and $y_i < y_j < y_i' < y_j'$ (or vice-versa).
Then $x_i < x_j'$ for all $i,j$ and $y_i < y_j'$ for all $i,j$. So $[\mathrm{max} (x_i) , \mathrm{min} (x_i')] \times [\mathrm{max} (y_i) , \mathrm{min}( y_i')]$ is a rectangle that works.
Here, it wasn't hard to find that intersection point with even without the reduction from Helly's theorem.
What kind of interesting collections of convex sets result in non-trivial uses of Helly's theorem?
