Almost Hadamard matrices: the case of arbitrary exponents

In our previous work, we introduced the following relaxation of the Hadamard property: a square matrix $H\in M_N(\mathbb{R})$ is called “almost Hadamard” if $U=H\mathrm{/}\sqrt{N}$ is orthogonal, and locally maximizes the 1-norm on O(N). We review our previous results, notably with the formulation of a new question, regarding the circulant and symmetric case. We discuss then an extension of the almost Hadamard matrix formalism, by making use of the p-norm on O(N), with $p\in [1,\mathrm{\infty }]-2$, with a number of theoretical results on the subject, and the formulation of some open problems.