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Convex Cylinders and the Symmetric Gaussian Isoperimetric Problem

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.