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Using the calculus of variations, we prove the following structure theorem for noise stable partitions: a partition of n-dimensional Euclidean space into m disjoint sets of fixed Gaussian volumes that maximize their noise stability must be (m − 1)-dimensional, if m − 1 ≤ n. In particular, the maximum noise stability of a partition of m sets in R n of fixed Gaussian volumes is constant for all n satisfying n ≥ m − 1. From this result, we obtain:(i) A proof of the Plurality is Stablest Conjecture for 3 candidate elections, for all correlation parameters ρ satisfying 0 < ρ < ρ 0 , where ρ 0 > 0 is a fixed constant (that does not depend on the dimension n), when each candidate has an equal chance of winning. (ii) A variational proof of Borell's Inequality (corresponding to the case m = 2). The structure theorem answers a question of De-Mossel-Neeman and of Ghazi-Kamath-Raghavendra. Item (i) is the first proof of any case of the Plurality is Stablest Conjecture of Khot-Kindler-Mossel-O'Donnell (2005) for fixed ρ, with the case ρ → 1 − being solved recently. Item (i) is also the first evidence for the optimality of the Frieze-Jerrum semidefinite program for solving MAX-3-CUT, assuming the Unique Games Conjecture. Without the assumption that each candidate has an equal chance of winning in (i), the Plurality is Stablest Conjecture is known to be false.

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