Convex Cylinders and the Symmetric Gaussian Isoperimetric Problem

Abstract

Let $\Omega$ be a measurable Euclidean set in ${\mathbb{R}}^n$ that is symmetric, i.e. $\Omega =-\Omega$, such that $\Omega \times \mathbb{R}$ has the smallest Gaussian surface area among all measurable symmetric sets of fixed Gaussian volume. We conclude that either $\Omega$ or ${\Omega }^c$ is convex. Moreover, except for the case $H(x)=⟨x,N(x)⟩+\lambda$ with $H\ge 0$ and $\lambda <0$, we show there exist a radius $r>0$ and an integer $0\le k\le n-1$ such that after applying a rotation, the boundary of $\Omega$ must satisfy $\partial \Omega =r{S}^k\times {\mathbb{R}}^{n-k-1}$, with $\sqrt{n-1}\le r\le \sqrt{n+1}$ when $k\ge 1$. Here $S^k$ denotes the unit sphere of ${\mathbb{R}}^{k+1}$ centered at the origin, and $n\ge 1$ is an integer. One might say this result nearly resolves the symmetric Gaussian conjecture of Barthe from 2001.

26 pages, 2 figures. arXiv admin note: text overlap with arXiv:1705.06643

Cited by in corpus (1)

• A remark on a conjecture on the symmetric Gaussian Problem