The Lp analogues of the Petty projection inequality and the Busemann-Petty centroid inequality are established.An affine isoperimetric inequality compares two functionals associated with convex (or more general) bodies, where the ratio of the functionals is invariant under non-degenerate linear transformations. These isoperimetric inequalities are more powerful than their better-known Euclidean relatives.This article deals with affine isoperimetric inequalities for centroid and projection bodies. Centroid bodies were attributed by Blaschke to Dupin (see e.g., the books of Schneider [32] and Leichtweiß [17] for references). If K is an origin-symmetric convex body in Euclidean nspace, R n , then the centroid body of K is the body whose boundary consists of the locus of the centroids of the halves of K formed when K is cut by codimension 1 subspaces. Blaschke (see Schneider [32] for references) conjectured that the ratio of the volume of a body to that of its centroid body attains its maximum precisely for ellipsoids. This conjecture was proven by Petty [27] who also extended the definition of centroid bodies and gave centroid bodies their name. When written as an inequality, Blaschke's conjecture is known as the Busemann-Petty centroid inequality. Busemann's name is attached to the inequality because Petty showed that Busemann's random simplex inequality ([5]) could be reinterpreted as what would become known as the Busemann-Petty centroid inequality. In recent times, centroid bodies (and their

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Abstract. In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite dimensional Banach space.

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