Given some unit vectors a1, ..., am ? Rn that span all Rn and some positive numbers q1, ..., qm, we consider for every p = 1 the convex body

Kp:={ x ? Rn : ?i=1m |<x, [(ai)/(qi)]>|p = 1}.

We will give some upper bounds for the volume of Kp and some lower bounds for the volume of its polar, depending on some parameters, which improve the ones obtained using the Brascamp-Lieb inequality. We will also see how the best choice of this parameters is related to the transformation which takes Kp to a special position which, for instance, when p=8, is John's position.