On Moment-Entropy inequalities in the space of matrices

In a series of works, Lutwak, Yang and Zhang established what could be called affine information theory, which is the study of moment-entropy and Fisher-information-type inequalities that are invariant with respect to affine transformations for random vectors. Their set of tools stemmed from sharp affine isoperimetric inequalities in the Lᵖ Brunn-Minkowski theory of convex geometry they had established. In this work, we generalize the affine information theory to the setting of matrices. These inequalities on the space of n× m matrices are induced by the interaction between Rⁿ with its Euclidean structure and Rᵐ equipped with a pseudo-norm.