For $\Omega\subset\mathbb{R}^n$ open and bounded, is it always the case that $\text{diam}(\Omega)$ is greater or equal to at least one side of the minimal rectangular box containing $\Omega$ ? Added : the sides of the minimal rectangular box are required to be parallel to the axes.
It looks true, but I can't prove it.
For a circle the minimal rectangular box would be a square and we would have equality.
For a square (with sides parallel to the axes), the minimal rectangular box would be the square itself and we would have a strict inequality, as the diagonal of a square is strictly greater in length than its sides.