A periodic isoperimetric problem related to the unique games conjecture

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 ≥ 2. Suppose a subset Ω of n‐dimensional Euclidean space double-struckRn satisfies −Ω = Ωc and Ω + v = Ωc (up to measure zero sets) for every standard basis vector v∈double-struckRn. For any x=false(x1,…,xnfalse)∈double-struckRn and for any q ≥ 1, let false‖xfalse‖qq=false|x1false|q+…+false|xnfalse|q and let γnfalse(xfalse)=false(2πfalse)−nfalse/2e−false‖xfalse‖22false/2 . For any x ∈ ∂Ω, let N(x) denote the exterior normal vector at x such that ‖N(x)‖2 = 1. Let B=false{x∈double-struckRn:sinfalse(πfalse(x1+…+xnfalse)false)≥0false}. Our main result shows that B has the smallest Gaussian surface area among all such subsets Ω, less a small error: ∫∂normalΩγnfalse(xfalse)normaldx≥false(1−6×10−9false)∫∂Bγnfalse(xfalse)normaldx+∫∂normalΩtrue(1−false‖Nfalse(xfalse)false‖1ntrue)γnfalse(xfalse)normaldx. In particular, ∫∂normalΩγnfalse(xfalse)normaldx≥false(1−6×10−9false)∫∂Bγnfalse(xfalse)normaldx. Standard arguments extend these results to a corresponding weak inequality for noise stability. Removing the factor 6 × 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.