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We prove that if a convex body has absolutely continuous surface areameasure, whose density is sufficiently close to the constant, then the sequence$\{\Pi^mK\}$ of convex bodies converges to the ball with respect to theBanach-Mazur distance, as $m\rightarrow\infty$. Here, $\Pi$ denotes theprojection body operator. Our result allows us to show that the ellipsoid is alocal solution to the conjectured inequality of Petty and to improve a relatedinequality of Lutwak.

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