A Periodic Isoperimetric Problem Related to the Unique Games Conjecture

Steven Heilman

Abstract:

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 $v\in\mathbb{R}^{n}$. We show that a ``periodic half space'' $B$ has the smallest Gaussian surface area among all such subsets $\Omega$, less a small error. Standard arguments extend these results to a corresponding weak inequality for noise stability. Removing the small error $6\cdot 10^{-9}$ 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.