Let $R$ be a collection of closed rectangles in $\mathbb{R}^2$ with sides parallel to the axes. Show that if any two rectangles in $R$ intersect, then all rectangles in $R$ have a common point.
Helly’s theorem requires any $d+1$ rectangles (in this case, 3 rectangles) to have non-empty intersection. However , this question only talks about the intersection of two rectangles as a condition. So, I don’t really know how to go about this.
Your help would be greatly appreciated.
On
Let us define four functions $\ell, r, b, t : R \to \mathbb{R}$ which stand for left, right, bottom, top respectively. i.e. $\ell([a,b]\times[c,d]) = a$, etc. We then claim that under the given conditions, $\sup_{x \in R} \ell(x) \le \inf_{x\in R} r(x)$ and $\sup_{x\in R} b(x) \le \inf_{x\in R} t(x)$. Therefore, we will have that $(\sup_{x\in R} \ell(x), \sup_{x\in R} b(x)) \in \bigcap R$.
To show the first inequality, suppose to the contrary that we had $\sup_{x\in R} \ell(x) > \inf_{x\in R} r(x)$. Then setting $m$ to be the midpoint between these two values, we can find $x \in R$ with $\ell(x) > m$, and we can also find $y \in R$ with $r(y) < m$. However, since then $r(y) < \ell(x)$, we must have that $x \cap y = \emptyset$, giving a contradiction. The proof that $\sup_{x\in R} b(x) \le \inf_{x\in R} t(x)$ is similar.
Take $3$ rectangles of $R$, $\left[a_1, b_1\right]\times \left[c_1, d_1\right]$, $\left[a_2, b_2\right]\times \left[c_2, d_2\right]$ and $\left[a_3, b_3\right]\times \left[c_2, d_2\right]$. Since the third rectangle intersects each of the two rectangles then $a_3\le b_1$, $a_3\le b_2$ and also $c_3\le d_1$ and $c_3\le d_2$. Use the same idea for the other rectangles you will have that $\left[a, b\right]\times \left[c, d\right]$ is in all three rectangles, where $a = \max\left\{a_1, a_2, a_3\right\}$, $b = \min\left\{b_1, b_2, b_3\right\}$, $c = \max\left\{c_1, c_2, c_3\right\}$ and $d = \max\left\{d_1, d_2, d_3\right\}$