Almost Hadamard matrices: general theory and examples

We develop a general theory of "almost Hadamard matrices". These are by definition the matrices $H\in M_N(\mathbb R)$ having the property that $U=H/\sqrt{N}$ is orthogonal, and is a local maximum of the 1-norm on O(N). Our study includes a detailed discussion of the circulant case ($H_{ij}=\gamma_{j-i}$) and of the two-entry case ($H_{ij}\in{x,y}$), with the construction of several families of examples, and some 1-norm computations.