In this note, we study the maximal perimeter of a convex set in ℝn with respect to various classes of measures. Firstly, we show that for a probability measure μ on ℝn, satisfying very mild assumptions, there exists a convex set of μ-perimeter at least C n Var|X|4𝔼|X|. This implies, in particular, that for any isotropic log-concave measure μ, one may find a convex set of μ-perimeter of order n1 8. Secondly, we derive a general upper bound of Cn||f||∞1 n on the maximal perimeter of a convex set with respect to any log-concave measure with density f in an appropriate position. Our lower bound is attained for a class of distributions including the standard normal distribution. Our upper bound is attained, say, for a uniform measure on the cube. In addition, for isotropic log-concave measures, we prove an upper bound of order n2 for the maximal μ-perimeter of a convex set.

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关于概率测度的凸集最大周长的一些评论

在这篇笔记中，我们研究了凸集的最大周长ℝn关于各类措施。首先，我们证明对于概率测度μ在ℝn，满足非常温和的假设，存在凸集μ- 至少周长C n 变量|X|4𝔼|X|.这尤其意味着，对于任何各向同性对数凹测量μ, 可以找到一组凸集μ- 订单周长n1 8. 其次，我们推导出一个一般的上界Cn||F||∞1 n关于任何具有密度的对数凹测度的凸集的最大周长F在适当的位置。我们的下限是针对包括标准正态分布在内的一类分布。比如说，对于立方体上的统一度量，我们的上限已经达到。此外，对于各向同性对数凹测量，我们证明了阶数的上限n2为最大μ- 凸集的周长。