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

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• 作者：C. Saroglou ^{csaroglo@kent.edu" class="auth_mail" title="E-mail the corresponding author christos.saroglou@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; A. Zvavitch ; ¹ ; ^{zvavitch@math.kent.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：52A20 ; 53A15 ; 52A39
• 刊名：Journal of Functional Analysis
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：272
• 期：2
• 页码：613-630
• 全文大小：382 K

文摘

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 {Π^mK}$\{{\mathrm{\Pi }}^{m}K\}$ of convex bodies converges to the ball with respect to the Banach–Mazur distance, as afc1857cc66ae99e83090" title="Click to view the MathML source">m→∞$m\to \infty$. 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.