A Rogers--Brascamp--Lieb--Luttinger inequality in the space of matrices

We consider convex bodies in $M_{n,m}(\mathbb{R})$, the space of matrices of $n$-rows and $m$-columns. A special case of fiber-symmetrization in $M_{n,m}(\mathbb{R})$ was recently introduced in [5,6]. We prove a Rogers–Brascamp–Lieb–Luttinger type inequality with respect to this symmetrization, for quasi-concave functions and provide some applications.