I came across the following while doing some related proof; It seems easy to prove $\newline$ $1$) An $n\times n$ matrix $U$ that is unitary has up to permuting columns a diagonal such the modulus of each entry $\le \dfrac{\sqrt{2}}{2}$ $2$) There are no $n\times n$ unitaries with constant diagonal $cI$ where $|c|>\dfrac{1}{\sqrt{n-1}}$. I am searching for a proof or related facts Thanks.