The Petty projection inequality for sets of finite perimeter

Within the Brunn-Minkowski theory, the two classical inequalities which connect the volume of a convex body with that of its polar projection body are the Petty and Zhang projection inequalities. Unlike the classical isoperimetric inequality (see, e.g., [19–21, 52]), the Petty and Zhang projection inequalities are affine isoperimetric inequalities in that they are inequalities between a pair of geometric functionals whose product is invariant under affine transformations. Many important results about affine isoperimetric inequalities and their functional forms have been found (see, e.g., [3, 7, 10, 17, 28–31, 43]). The Petty projection inequality strengthens and directly implies the classical isoperimetric inequality, but it can be viewed as an optimal isoperimetric inequality. It is the geometric core of the affine Sobolev-Zhang inequality [61] which strengthens the classical sharp Euclidean Sobolev inequality. The Lp version of Petty’s inequality by Lutwak et al. [43] and its Orlicz extension by the same authors [48] both represent landmark results in the evolution of the Brunn-Minkowski theory first towards an Lp theory and, more recently, towards an Orlicz theory of convex bodies. The Lp and Orlicz theories of convex bodies have expanded rapidly (see, e.g., [7, 12, 22, 23, 26, 27, 32, 34–42, 44–47, 60, 62]). Affine isoperimetric inequalities referring to convex bodies have been extended to certain classes of non-convex domains such as star bodies and sets of finite perimeter. The Petty projection inequality has