This is an announcement for the paper "Iterations of the projection body operator and a remark on Petty's conjectured projection inequality" by Christos Saroglou and Artem Zvavitch.

Abstract: We prove that if a convex body has absolutely continuous surface area measure, whose density is sufficiently close to the constant, then the sequence ${\Pi^mK}$ of convex bodies converges to the ball with respect to the Banach-Mazur distance, as $m\rightarrow\infty$. Here, $\Pi$ 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.

Archive classification: math.MG math.FA

Remarks: 13 pages

Submitted from: csaroglo@kent.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1511.03381

or

http://arXiv.org/abs/1511.03381