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 ≥ 2. Suppose a subset Ω of n-dimensional Euclidean space R satisfies −Ω = Ω and Ω + v = Ω for any standard basis vector v ∈ R. For any x = (x1, . . . , xn) ∈ R and for any q ≥ 1, let ‖x‖q = |x1| + · · · + |xn| and let γn(x) = (2π)−n/2e−‖x‖ 2 2/2 . For any x ∈ ∂Ω, let N(x) denote the exterior normal vector at x such that ‖N(x)‖2 = 1. Let B = {x ∈ R : sin(π(x1 + · · · + xn)) ≥ 0}. Our main result shows that B has the smallest Gaussian surface area among all such subsets Ω, less a small error: ∫ ∂Ω γn(x)dx ≥ (1− 6 · 10−9) ∫