Maximal Surface Area of a Convex Set in $$\mathbb{R}^{n}$$ with Respect to Log Concave Rotation Invariant Measures

Abstract

AbstractIt 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.KeywordsMaximum Surface AreaRotation Invariant MeasureStandard Gaussian MeasureNazarovSymmetry MeasureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.