On orthogonal matrices maximizing the 1-norm

For $U\in O(N)$ we have $\mathrm{\mid }\mathrm{\mid }U\mathrm{\mid }{\mathrm{\mid }}_1\le N\sqrt{N}$, with equality if and only if $U=H\mathrm{/}\sqrt{N}$, with $H$ Hadamard matrix. Motivated by this remark, we discuss in this paper the algebraic and analytic aspects of the computation of the maximum of the 1-norm on O(N). The main problem is to compute the $k$-th moment of the 1-norm, with $k\to \mathrm{\infty }$, and we present a number of general comments in this direction.