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.
The article deals with affine isoperimetric inequalities for centroid and projection bodies. The most important inequality concerning centroid bodies is the Busemann-Petty centroid inequality that answers positively a conjecture by Blaschke proving that the ratio of the volume of a body to that of its centroid body attains its maximum precisely for ellipsoids. The fundamental inequality for projection bodies is the Petty projection inequality: of all convex bodies of fixed volume, the ones whose polar projection bodies have maximal value are precisely the ellipsoids. Both the Petty projection inequality and the Busemann-Petty centroid inequality have come to be recognized as fundamental affine inequalities and have attracted increased attention.
In this paper the authors derive the exact \(L_p\)-analogues of both the Busemann-Petty centroid inequality and the Petty projection inequality (as well as their equality conditions). The inequality concerning the \(L_p\)-centroid body is stronger than the well-known Blaschke-Santaló’s inequality.