Mean projection and section radii of convex bodies

We introduce new series of mean outer and inner radii, which are defined as the outer (respectively, inner) radius of, either the projection of the convex body onto an i -dimensional subspace, or the i -dimensional section, 1 ≤ i ≤ n , averaged over the Grassmannian manifold, and with respect to the Haar probability measure. We study some properties of these new functionals, establishing inequalities among them, as well as their relation with other measures as the volume or the quermassintegrals.