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We prove the endpoint case of a conjecture of Khot and Moshkovitz related tothe 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 $\Omega+v=\Omega^{c}$ (upto measure zero sets) for every standard basis vector $v\in\mathbb{R}^{n}$. Forany $x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$ 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 $x$ suchthat $\|N(x)\|_{2}=1$. Let $B=\{x\in\mathbb{R}^{n}\colon\sin(\pi(x_{1}+\cdots+x_{n}))\geq0\}$. Our main result shows that $B$ has thesmallest Gaussian surface area among all such subsets $\Omega$, less a smallerror: $$ \int_{\partial\Omega}\gamma_{n}(x)dx\geq(1-6\cdot10^{-9})\int_{\partialB}\gamma_{n}(x)dx+\int_{\partial\Omega}\Big(1-\frac{\|N(x)\|_{1}}{\sqrt{n}}\Big)\gamma_{n}(x)dx.$$ In particular, $$ \int_{\partial\Omega}\gamma_{n}(x)dx\geq(1-6\cdot10^{-9})\int_{\partial B}\gamma_{n}(x)dx. $$ Standard arguments extend theseresults to a corresponding weak inequality for noise stability. Removing thefactor $6\cdot 10^{-9}$ would prove the endpoint case of the Khot-Moshkovitzconjecture. Lastly, we prove a Euclidean analogue of the Khot and Moshkovitzconjecture.  The full conjecture of Khot and Moshkovitz provides strong evidence for thetruth of the Unique Games Conjecture, a central conjecture in theoreticalcomputer science that is closely related to the P versus NP problem. So, ourresults also provide evidence for the truth of the Unique Games Conjecture.Nevertheless, this paper does not prove any case of the Unique Gamesconjecture.

A Periodic Isoperimetric Problem Related to the Unique Games Conjecture