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Affine inequalities for Lp mean zonoids

In this paper, the Lp ( p⩾1 ) mean zonoid of a convex body K is given, and we show that it is the Lp centroid body of radial (n+p) th mean body of K up to a dilation. We also establish some affine inequalities of these bodies by proving that the volume of the new bodies is decreasing under Steiner symmetrization.