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

2015/11/11 by Christos Saroglou, Saroglou, Christos, Artem Zvavitch +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #math.FA #math.MG

paper · pdf · doi:10.48550/arxiv.1511.03381

13 pages

arxiv created 2015/11/11 · arxiv updated 2015/11/12

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 \Π^mK\ 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.

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