Almost Hadamard matrices: the case of arbitrary exponents

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In our previous work, we introduced the following relaxation of the Hadamard property: a square matrix H∈ Mₙ(R) is called "almost Hadamard" if U=H/√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∈[1,∞]-2, with a number of theoretical results on the subject, and the formulation of some open problems.