A Periodic Isoperimetric Problem Related to the Unique Games Conjecture

We prove the endpoint case of a conjecture of Khot and Moshkovitz related to the Unique Games Conjecture, less a small error. Let $n\geq2$. Suppose a subset $\Omega$ of $n$-dimensional Euclidean space $\mathbb{R}^{n}$ satisfies $-\Omega=\Omega^{c}$ and $\Omega+v=\Omega^{c}$ (up to measure zero sets) for every standard basis vector $v\in\mathbb{R}^{n}$. For any $x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$ and for any $q\geq1$, let $\|x\|_{q}^{q}=|x_{1}|^{q}+\cdots+|x_{n}|^{q}$ and let $\gamma_{n}(x)=(2\pi)^{-n/2}e^{-\|x\|_{2}^{2}/2}$ . For any $x\in\partial\Omega$, let $N(x)$ denote the exterior normal vector at $x$ such that $\|N(x)\|_{2}=1$. Let $B=\{x\in\mathbb{R}^{n}\colon \sin(\pi(x_{1}+\cdots+x_{n}))\geq0\}$. Our main result shows that $B$ has the smallest Gaussian surface area among all such subsets $\Omega$, less a small error: $$\int_{\partial\Omega}\gamma_{n}(x)dx\geq(1-6\cdot 10^{-9})\int_{\partial B}\gamma_{n}(x)dx+\int_{\partial\Omega}\Big(1-\frac{\|N(x)\|_{1}}{\sqrt{n}}\Big)\gamma_{n}(x)dx.$$ In particular, $$\int_{\partial\Omega}\gamma_{n}(x)dx\geq(1-6\cdot 10^{-9})\int_{\partial B}\gamma_{n}(x)dx.$$ Standard arguments extend these results to a corresponding weak inequality for noise stability. Removing the factor $6\cdot 10^{-9}$ would prove the endpoint case of the Khot-Moshkovitz conjecture. Lastly, we prove a Euclidean analogue of the Khot and Moshkovitz conjecture. The full conjecture of Khot and Moshkovitz provides strong evidence for the truth of the Unique Games Conjecture, a central conjecture in theoretical computer science that is closely related to the P versus NP problem. So, our results also provide evidence for the truth of the Unique Games Conjecture. Nevertheless, this paper does not prove any case of the Unique Games conjecture.