Problem 1471

1. Projection onto a Plane:
Consider any plane $\pi$ in $\mathbb{R}^3$. For each convex subset $K \subseteq \mathbb{R}^3$, project $K$ onto the plane $\pi$. Denote the projection of $K$ onto $\pi$ by $P(K)$. Since $K$ is convex, $P(K)$ is also a convex set in the plane $\pi$.

2. Application of Helly's Theorem in the Plane:
By the problem's condition, every three of the original convex subsets in $\mathbb{R}^3$ have a non-empty intersection. This property is preserved under projection. Therefore, every three of the projected convex sets $P(K)$ in the plane $\pi$ also have a non-empty intersection.

Helly's theorem states that for a finite collection of at least $d+1$ convex sets in $\mathbb{R}^d$, if the intersection of every $d+1$ of these sets is non-empty, then the intersection of all the sets is non-empty. In our case, $d=2$ (since we are in the plane $\pi$), and every three (i.e., $d+1$) of the projected convex sets have a non-empty intersection. Therefore, by Helly's theorem, the intersection of all the projected convex sets $P(K)$ is non-empty. Let $p \in \pi$ be a point in this intersection.

3. Constructing the Stabbing Line:
Consider the line $L$ in $\mathbb{R}^3$ that passes through the point $p$ and is normal to the plane $\pi$. This line $L$ intersects the plane $\pi$ at the point $p$.

4. Intersection with Original Convex Sets:
Since $p$ is in the intersection of all the projections $P(K)$, the line $L$ intersects each original convex set $K$ at some point. This is because the projection of $K$ onto $\pi$ contains $p$, and thus $K$ must intersect the line $L$ at some point above or below $p$ in $\mathbb{R}^3$.

5. Generalization to Higher Dimensions:
The argument can be generalized to any dimension $n$. If we have finitely many convex subsets in $\mathbb{R}^n$ such that every $n$ of them have a non-empty intersection, we can project these sets onto an $(n-1)$-dimensional hyperplane and apply Helly's theorem in $\mathbb{R}^{n-1}$. The same reasoning shows that there exists a line in $\mathbb{R}^n$ that intersects all the convex subsets.

Therefore, we have shown that there exists a line in $\mathbb{R}^3$ that intersects all the given convex subsets.

$\blacksquare$