Gram determinants of real binary tensors

A binary tensor consists of 2n entries arranged into hypercube format 2×2×⋯×2. There are n ways to flatten such a tensor into a matrix of size 2×2n−1. For each flattening, M, we take the determinant of its Gram matrix, det(MMT). We consider the map that sends a tensor to its n-tuple of Gram determinants. We propose a semi-algebraic characterization of the image of this map. This offers an answer to a question raised by Hackbusch and Uschmajew concerning the higher-order singular values of tensors.