Lp AFFINE ISOPERIMETRIC INEQUALITIES

Affine isoperimetric inequalities compare functionals, associated with convex (or more general) bodies, whose ratios are invariant under GL(n)-transformations of the bodies. These isoperimetric inequalities are more powerful than their better-known relatives of a Euclidean flavor. To be a bit more specific, this article deals with inequalities for centroid and projection bodies. Centroid bodies were attributed by Blaschke (see e.g., the books of Schneider [S2] and Leichtweiß [Le] for references) to Dupin. If K is an origin-symmetric convex body in Euclidean n-space, R, 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 [S2] 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 [P1] 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 ([Bu]) could be reinterpreted as what would become known as the Busemann-Petty centroid inequality. In recent times, centroid bodies (and their associated inequalities) have attracted increased attention (see e.g. Milman and Pajor [MPa1,MPa2]). In retrospect, it can be seen that much if not all of this recent interest was inspired by Petty’s seminal work [P1]. Projection bodies are of newer vintage. They were introduced at the turn of the previous century by Minkowski. He showed that corresponding to each convex body K in R is a unique origin-symmetric convex body ΠK, the projection body of K, which can be defined (up to dilation) by the amazing fact that the following ratio is