Iterations of the projection body operator and a remark on Petty's conjectured projection inequality

We prove that if a convex body has an absolutely continuous surface area measure, whose density is sufficiently close to a constant function, then the sequence {ΠmK} of convex bodies converges to the ball with respect to the Banach-Mazur distance, as m→∞. Here, Π denotes the projection body operator. Our result allows us to show that the ellipsoid is a local solution to the conjectured inequality of Petty and to improve a related inequality of Lutwak.