论文标题
高斯空间中的尺寸Brunn-Minkowski不平等
The dimensional Brunn-Minkowski inequality in Gauss space
论文作者
论文摘要
令$γ_n$为$ \ mathbb {r}^n $上的标准高斯度量。我们证明，对于每一个对称凸集，$ k，l $ in $ \ mathbb {r}^n $和（0,1）$，$$γ_n（λk+（1-λ）^{\ frac {\ frac {1}} {n}} {n}} \ geq λγ_n（k）^{\ frac {1} {n}}+（1-λ）γ_n（l）^{\ frac {1} {n}}，$$，因此解决了Gardner and Zvavitch（2010）提出的问题。这是Lebesgue度量的经典Brunn-Minkowski不平等的高斯类似物。我们还表明，对于固定的$λ\在（0,1）$中，当且仅当$ k = l $时达到平等。
Let $γ_n$ be the standard Gaussian measure on $\mathbb{R}^n$. We prove that for every symmetric convex sets $K,L$ in $\mathbb{R}^n$ and every $λ\in(0,1)$, $$γ_n(λK+(1-λ)L)^{\frac{1}{n}} \geq λγ_n(K)^{\frac{1}{n}}+(1-λ)γ_n(L)^{\frac{1}{n}},$$ thus settling a problem raised by Gardner and Zvavitch (2010). This is the Gaussian analogue of the classical Brunn-Minkowski inequality for the Lebesgue measure. We also show that, for a fixed $λ\in(0,1)$, equality is attained if and only if $K=L$.
