  We reformulate, modify and extend a comparison criteria of $L^{p}$ norms
obtained by Nazarov-Podkorytov and place it in the general setting of
interpolation theory and majorization theory. In particular, we give norm
comparison criteria for general scales of interpolation spaces, including
non-commutative $L^{p}$ and Lorentz spaces. As an application, we extend the
classical Ball's integral inequality, which lies at the basis of his famous
result on sections of the $n-$dimensional unit cube.
