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

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We prove that if a convex body has absolutely continuous surface area measure, whose density is sufficiently close to the constant, 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.