## Rectangles in the plane

We consider a finite family of rectangles in plane, having edges parallel with the coordinate axes such that for any pair of rectangles there exists a vertical line, or a horizontal one which intersects both rectangles. Prove that there exists a vertical line and a horizontal one such that any of the given rectangles intersects at least one of them.

**Proof: **If no pair of rectangles can be joined by a vertical line then every two rectangles can be joined by a horizontal line. By Helly’s theorem (one dimensional case), the projections of the rectangles on the axis have a common point, which means that there exists a horizontal line joining all the rectangles. This line paired with any vertical line solve our problem.

Next, suppose that we have at least two rectangles which can be joined by a vertical line. Denote a family of rectangles such that any two of them can be joined by a vertical line. Considering the projections of the rectangles in on the axis and applying Helly’s theorem again, we see that the intersection of the projection is a non-void and compact interval . Denote by the family formed by the rest of rectangles. If , then we are done. Else we split in two sets as being the rectangles whose projections on are on the left of , and respectively on the right. Since the intersection yielding is finite, there exists on rectangle whose right vertical side projects on the right frontier of . This means that all the rectangles in (if there are some) cannot be joined verticaly with , and therefore are all joined horizontally with . We do the same thing to find a similar rectangle which intersects horizontally every rectangle in .

To be continued…