  Let $\Omega\subset\mathbb{R}^{n+1}$ have minimal Gaussian surface area among
all sets satisfying $\Omega=-\Omega$ with fixed Gaussian volume. Let $A=A_{x}$
be the second fundamental form of $\partial\Omega$ at $x$, i.e. $A$ is the
matrix of first order partial derivatives of the unit normal vector at
$x\in\partial\Omega$. For any $x=(x_{1},\ldots,x_{n+1})\in\mathbb{R}^{n+1}$,
let $\gamma_{n}(x)=(2\pi)^{-n/2}e^{-(x_{1}^{2}+\cdots+x_{n+1}^{2})/2}$. Let
$\|A\|^{2}$ be the sum of the squares of the entries of $A$, and let
$\|A\|_{2\to 2}$ denote the $\ell_{2}$ operator norm of $A$.
  It is shown that if $\Omega$ or $\Omega^{c}$ is convex, and if either
$$\int_{\partial\Omega}(\|A_{x}\|^{2}-1)\gamma_{n}(x)dx>0\qquad\mbox{or}\qquad
\int_{\partial\Omega}\Big(\|A_{x}\|^{2}-1+2\sup_{y\in\partial\Omega}\|A_{y}\|_{2\to
2}^{2}\Big)\gamma_{n}(x)dx<0,$$ then $\partial\Omega$ must be a round cylinder.
That is, except for the case that the average value of $\|A\|^{2}$ is slightly
less than $1$, we resolve the convex case of a question of Barthe from 2001.
  The main tool is the Colding-Minicozzi theory for Gaussian minimal surfaces,
which studies eigenfunctions of the Ornstein-Uhlenbeck type operator $L=
\Delta-\langle x,\nabla \rangle+\|A\|^{2}+1$ associated to the surface
$\partial\Omega$. A key new ingredient is the use of a randomly chosen degree 2
polynomial in the second variation formula for the Gaussian surface area. Our
actual results are a bit more general than the above statement. Also, some of
our results hold without the assumption of convexity.
