A diagonal $\overline{uv}$ in simple polygon $P$ is called short if the distance from $u$ to $v$ is 2 in $P$ (two segments between them). Prove or disprove that every simple polygon has a short diagonal.
I always have a problem with the proofs like that, it's hard to prove obvious things, and very often it's very misleading, what seems obvious for me actually cannot be true any more in some extreme case.
If you have any idea about the proof, please, share it with us. Thanks!