A question about drawing a line throught n connected, pairwise intersecting regions.

Question

A friend asked me this: Given n connected regions in space, and any pair of regions has no empty intersection, is it possible to draw a line through all of the regions?

Answer 1

HINT

Not only is a line through all regions possible, you can in fact choose any orientation you like and find a line of that orientation through all regions.

E.g. how about a line parallel to the $y$-axis? Such a line is defined by $x=c$ for some constant $c$. For region $A$, let $x(A)$ denote the set of $x$ values contained in the region, i.e.,

$$x(A) = \{x \in \mathbb{R} \mid \exists y: (x,y) \in A\}$$

So the line $x=c$ intersects $A$ iff $c \in x(A)$. This being a puzzle, I don't want to give the whole solution away, so...

• Each region is connected. What does that say about $x(A)$?

• Every pair of regions $A,B$ overlaps. What does that say about $x(A), x(B)$?

Can you finish from here?