Let $K$ be the set of $n\times n$ matrices with zero diagonal entries and such that the sum of all entries is zero.

The cut norm of a $n\times n$ matrix $M$ is:

$$ cut(M) = \sup_{S, T, S\cap T = \emptyset} \left \vert \sum_{i\in S, j\in T} M_{ij}\right \vert $$

How large can $\sum_{ij}\vert M_{ij}\vert$ be for a matrix $M$ in $K$ with cut norm $1$?

Is it possible to build explicit matrices attaining a high value?

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