Inequalities for dual isoperimetric deficits

We study dual isoperimetric deficits of star bodies. We introduce the dual Steiner ball of a star body, and use it to establish an inequality for the L-p distance, p = 2 and p = infinity, between the radial functions of two c onvex bodies. By applying this inequality, we find dual Bonnesen-type inequ alities for convex bodies. Finally, we use a general form of Gruss's inequa lity to derive dual Favard-type inequalities for star and convex bodies. Th e results contribute to the dual Brunn-Minkowski theory initiated by E. Lut wak, and continue the attempt to understand the relation between this and t he classical Brunn-Minkowski theory.