Stability Inequalities for Projections of Convex Bodies

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• 作者：Alexander Koldobsky
• 关键词：Convex body ; Projection ; Fourier transform ; Distribution
• 刊名：Discrete & Computational Geometry
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：57
• 期：1
• 页码：152-163
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Combinatorics; Computational Mathematics and Numerical Analysis;
• 出版者：Springer US
• ISSN：1432-0444
• 卷排序：57

文摘

The projection function \(P_K\) of an origin-symmetric convex body K in \({{\mathbb {R}}}^n\) is defined by \(P_K(\xi )=|K\vert {\xi ^\bot }|,\ \xi \in S^{n-1},\) where \(K\vert {\xi ^\bot }\) is the projection of K to the central hyperplane \(\xi ^\bot \) perpendicular to \(\xi \), and |K| stands for volume of proper dimension. We prove several stability and separation results for the projection function. For example, if D is a projection body in \({{\mathbb {R}}}^n\) which is in isotropic position up to a dilation, and K is any origin-symmetric convex body in \({{\mathbb {R}}}^n\) such that that there exists \(\xi \in S^{n-1}\) with \(P_K(\xi )>P_D(\xi ),\) then $$\begin{aligned} \max _{\xi \in S^{n-1}} (P_K(\xi )-P_D(\xi )) \ge \frac{c}{\log ^2n} \big (|K|^{\frac{n-1}{n}} -|D|^{\frac{n-1}{n}}\big ), \end{aligned}$$where c is an absolute constant. As a consequence, we prove a hyperplane inequality $$\begin{aligned} S(D) \le \ C \log ^2n \max _{\xi \in S^{n-1}} S(D\vert \xi ^\bot )\ |D|^{\frac{1}{n}}, \end{aligned}$$where D is a projection body in isotropic position, up to a dilation, S(D) is the surface area of \(D,S(D\vert \xi ^\bot )\) is the surface area of the body \(D\vert \xi ^\bot \) in \({{\mathbb {R}}}^{n-1},\) and C is an absolute constant. The proofs are based on the Fourier analytic approach to projections developed in [12].KeywordsConvex bodyProjectionFourier transformDistributionEditor in Charge: Günter M. Ziegler