球对称对数凹概率测度的等视野
Isoperimetry for spherically symmetric log-concave probability measures
摘要：
证明了概率测度$μ$在$\mathbb{R}^n$上的等周不等式，其中$μ$是$\mathbb{R}^n$上的欧氏范数，$ϕ$是非减凸函数.当$ϕ(x)=x^α$与$α\ge1$配合使用时，尤其适用。在对$ϕ$的温和假设下，如果选择$λ$使得$μ$的协方差是恒等式，则不等式是无量纲的。
英文摘要：
We prove an isoperimetric inequality for probability measures $μ$ on $\mathbb{R}^n$ with density proportional to $\exp(-ϕ(λ| x|))$, where $|x|$ is the euclidean norm on $\mathbb{R}^n$ and $ϕ$ is a non-decreasing convex function. It applies in particular when $ϕ(x)=x^α$ with $α\ge1$. Under mild assumptions on $ϕ$, the inequality is dimension-free if $λ$ is chosen such that the covariance of $μ$ is the identity.
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