Equivalence of Some Affine Isoperimetric Inequalities

Abstract

We establish the equivalence of some affine isoperimetric inequalities which include the -Petty projection inequality, the -Busemann-Petty centroid inequality, the "dual" -Petty projection inequality, and the "dual" -Busemann-Petty inequality. We also establish the equivalence of an affine isoperimetric inequality and its inclusion version for -John ellipsoids.