Finitely many convex subsets of $\mathbb R^3$ are given, such that every three have non-empty intersection. Prove that there exists a line in $\mathbb R^3$ that intersects all of these subsets.
Problem
Source: Iranian National Olympiad (3rd Round) 2004
Tags: geometry, geometry proposed
Radar
14.10.2014 00:39
Seems pretty similar to Helly's theorem...
mavropnevma
14.10.2014 17:00
It is. Consider any plane $\pi$. The projection of any convex set $K\subseteq \mathbb{R}^3$ on $\pi$ is a plane convex set. Therefore the family of the projections have the property of every three having non-empty intersection; thus, by Helly's theorem in the plane, the projections have a common point in $\pi$. The normal at that point to $\pi$ intersects all the original sets. Clearly, this may be generalized to any dimension $n$ where any $n$ of the convex sets have non-empty intersection. So we proved even more, namely that for any given direction there exists a stabbing line, parallel to that direction (take $\pi$ normal to the direction). For the plane, this was Problem 23 in [Hadwiger & Debrunner - Combinatorial Geometry in the Plane], with a different proof, that showed that such a line exists through any point in the plane. My method above shows such a line exists, parallel to any given direction in the plane. The book result follows from it via a continuity argument.