Maximal Surface Area of a Convex Set in \(\mathbb{R}^{n}\) with Respect to Log Concave Rotation Invariant Measures
Maximal Surface Area of a Convex Set in \(\mathbb{R}^{n}\) with Respect to Log Concave Rotation Invariant Measures
复制标题
(mathbb{R}^{n}) 中凸集的最大表面积相对于对数凹旋转不变测量
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
G. Livshyts
中科院分区：
文献类型：
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作者：
G. Livshyts
It was shown by K. Ball and F. Nazarov, that the maximal surface area of a convex set in \(\mathbb{R}^{n}\) with respect to the Standard Gaussian measure is of order \(n^{\frac{1} {4} }\). In the present paper we establish the analogous result for all rotation invariant log concave probability measures. We show that the maximal surface area with respect to such measures is of order \(\frac{\sqrt{n}} {\root{4}\of{\mathit{Var}\vert X\vert }\sqrt{\mathbb{E}\vert X\vert }}\), where X is a random vector in \(\mathbb{R}^{n}\) distributed with respect to the measure.
It was shown by K. Ball and F. Nazarov, that the maximal surface area of a convex set in \(\mathbb{R}^{n}\) with respect to the Standard Gaussian measure is of order \(n^{\frac{1} {4} }\). In the present paper we establish the analogous result for all rotation invariant log concave probability measures. We show that the maximal surface area with respect to such measures is of order \(\frac{\sqrt{n}} {\root{4}\of{\mathit{Var}\vert X\vert }\sqrt{\mathbb{E}\vert X\vert }}\), where X is a random vector in \(\mathbb{R}^{n}\) distributed with respect to the measure.