Almost Hadamard matrices: The case of arbitrary exponents

A square matrix H∈MN(R)H∈MN(R) is called “almost Hadamard” if U=H/N is orthogonal, and locally maximizes the 1-norm on O(N)O(N). We review our previous work on the subject, 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 pp-norm on O(N)O(N), with p∈[1,∞]−{2}p∈[1,∞]−{2}, with a number of theoretical results on the subject, and the formulation of some open problems.