关于Gardner-Zvavitch猜想：Brunn-Minkowski型不等式的对称性
On the Gardner-Zvavitch conjecture: symmetry in the inequalities of Brunn-Minkowski type
摘要：
本文研究Gardner和Zvavitch从\cite{GZ}中提出的猜想，证明了标准高斯测度$γ$关于\textbf{symmetric}凸集的Minkowski加法具有$\frac{1}{n}$ -可积性.我们证明了这一事实的一个因素的2：也就是说，我们表明，对称凸$K$和$L,$ $$ γ(λK+(1-λ)L)^{\frac{1}{2n}}\geq λγ(K)^{\frac{1}{2n}}+(1-λ)γ(L)^{\frac{1}{2n}}. $$此外，我们表明，在适当的无量纲的统一界限上的潜在的海森，甚至措施的对数可以加强到$p$，与$p>0,$相对于对称凸集的添加。
英文摘要：
In this paper, we study the conjecture of Gardner and Zvavitch from \cite{GZ}, which suggests that the standard Gaussian measure $γ$ enjoys $\frac{1}{n}$-concavity with respect to the Minkowski addition of \textbf{symmetric} convex sets. We prove this fact up to a factor of 2: that is, we show that for symmetric convex $K$ and $L,$ $$ γ(λK+(1-λ)L)^{\frac{1}{2n}}\geq λγ(K)^{\frac{1}{2n}}+(1-λ)γ(L)^{\frac{1}{2n}}. $$ Further, we show that under suitable dimension-free uniform bounds on the Hessian of the potential, the log-concavity of even measures can be strengthened to $p$-concavity, with $p>0,$ with respect to the addition of symmetric convex sets.
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