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Questions of noise stability play an important role in hardness of approximation in computer science as well as in the theory of voting. In many applications, the goal is to find an optimizer of noise stability among all possible partitions of Rn for n ≥ 1 to k parts with given Gaussian measures μ1, ..., μk. We call a partition e-optimal, if its noise stability is optimal up to an additive e. In this paper, we give an explicit, computable function n(e) such that an e-optimal partition exists in Rn(e). This result has implications for the computability of certain problems in non-interactive simulation, which are addressed in a subsequent work.

The Majority is Stablest Theorem has numerous applications in hardness of approximation and social choice theory. We give a new proof of the Majority is Stablest Theorem by induction on the dimension of the discrete cube. Unlike the previous proof, it uses neither the "invariance principle" nor Borell's result in Gaussian space. The new proof is general enough to include all previous variants of majority is stablest such as "it ain't over until it's over" and "Majority is most predictable". Moreover, the new proof allows us to derive a proof of Majority is Stablest in a constant level of the Sum of Squares hierarchy. This implies in particular that Khot-Vishnoi instance of Max-Cut does not provide a gap instance for the Lasserre hierarchy.