Isotropic surface area measures

The purpose of this note is to bring into attention an apparently forgotten result of C. M. Petty: a convex body has minimal surface area among its affine transformations of the same volume if, and only if, its area measure is isotropic. We obtain sharp affine inequalities which demonstrate the fact that this "surface isotropic" position is a natural framework for the study of hyperplane projections of convex bodies. §1. Introduction. We shall work in R" equipped with a fixed Euclidean structure and write | • | for the corresponding Euclidean norm. We denote the Euclidean unit ball and the unit sphere by Dn and S"~ ) respectively, and we write a for the rotationally invariant probability measure on S"~ l . The volume of appropriate dimension will be also denoted by | • |. We shall write con for the volume of the Euclidean unit ball in R". Finally, L(R", R") is the space of all linear transformations of R". Let K be a convex body in R". The area measure aK is denned on S"~ ) and corresponds to the usual surface measure on K via the Gauss map. If A is a Borel subset of S"~\ then