Three candidate plurality is stablest for small correlations

Abstract Using the calculus of variations, we prove following structure theorem for noise-stable partitions: a partition n -dimensional Euclidean space into m disjoint sets fixed Gaussian volumes that maximise their noise stability must be $(m-1)$ -dimensional, if $m-1\leq n$ . In particular, maximum in $\mathbb {R}^{n}$ is constant all satisfying $n\geq m-1$ From this result, obtain: (i) A proof plurality stablest conjecture three candidate elections, correlation parameters $\rho $ $0<\rho <\rho _{0}$ , where _{0}>0$ (that does not depend on dimension ), when each has an equal chance winning. (ii) variational Borell’s inequality (corresponding to case $m=2$ ). The answers question De–Mossel–Neeman and Ghazi–Kamath–Raghavendra. Item first any Khot-Kindler-Mossel-O’Donnell with \to L1^{-}$ being solved recently. also evidence optimality Frieze–Jerrum semidefinite program solving MAX-3-CUT, assuming unique games conjecture. Without assumption winning (i), known false.