论文标题
凸缸和对称的高斯等等问题
Convex Cylinders and the Symmetric Gaussian Isoperimetric Problem
论文作者
论文摘要
让$ω$为$ \ mathbb {r}^{n} $，是对称的，即$ω=-Ω$，以至于$ \ times \ times \ times \ times \ mathbb {r} $具有所有可测量的对称固定要的固定要量。我们得出结论，$ω$或$ω^{c} $是凸。此外，除了情况外，$ h（x）= \ langle x，n（x）\ rangle+λ$带有$ h \ geq0 $和$λ<0 $，我们显示出存在一个半径$ r> 0 $和整数$ 0 \ leq k \ leq k \ leq n-1 $ rs^{k} \ times \ mathbb {r}^{n-k-1} $，带有$ \ sqrt {n-1} \ leq r \ leq r \ leq \ leq \ sqrt {n+1} $时，$ k \ geq1 $。这里$ s^{k} $表示$ \ mathbb {r}^{k+1} $的单位球，以原点为中心，$ n \ geq1 $是整数。有人可能会说，从2001年起，这一结果几乎可以解决Barthe的对称高斯猜想。
Let $Ω$ be a measurable Euclidean set in $\mathbb{R}^{n}$ that is symmetric, i.e. $Ω=-Ω$, such that $Ω\times\mathbb{R}$ has the smallest Gaussian surface area among all measurable symmetric sets of fixed Gaussian volume. We conclude that either $Ω$ or $Ω^{c}$ is convex. Moreover, except for the case $H(x)=\langle x,N(x)\rangle+λ$ with $H\geq0$ and $λ<0$, we show there exist a radius $r>0$ and an integer $0\leq k\leq n-1$ such that after applying a rotation, the boundary of $Ω$ must satisfy $\partialΩ= rS^{k}\times\mathbb{R}^{n-k-1}$, with $\sqrt{n-1}\leq r\leq\sqrt{n+1}$ when $k\geq1$. Here $S^{k}$ denotes the unit sphere of $\mathbb{R}^{k+1}$ centered at the origin, and $n\geq1$ is an integer. One might say this result nearly resolves the symmetric Gaussian conjecture of Barthe from 2001.
