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 an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302257&_mathId=si1.gif&_user=111111111&_pii=S0022123616302257&_rdoc=1&_issn=00221236&md5=3807b0b3b315896b37164e2ac22f24eb" title="Click to view the MathML source">{Π^mK}an>an class="mathContainer hidden">an class="mathCode"><math altimg="si1.gif" overflow="scroll"><mo stretchy="false">{mo><msup><mrow><mi mathvariant="normal">Πmi>mrow><mrow><mi>mmi>mrow>msup><mi>Kmi><mo stretchy="false">}mo>math>an>an>an> of convex bodies converges to the ball with respect to the Banach&ndash;Mazur distance, as an id="mmlsi2" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302257&_mathId=si2.gif&_user=111111111&_pii=S0022123616302257&_rdoc=1&_issn=00221236&md5=80a0b208a67afc1857cc66ae99e83090" title="Click to view the MathML source">m&rarr;∞an>an class="mathContainer hidden">an class="mathCode"><math altimg="si2.gif" overflow="scroll"><mi>mmi><mo stretchy="false">&rarr;mo><mo>∞mo>math>an>an>an>. 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.