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In 1985 C. Borell proved that under the Gaussian measure, half-spaces are the most stable sets. While a number of proofs of this result were discovered over the years, it was not known if half-spaces are the unique optimizers. The talk will survey recent results with Joe Neeman establishing that half-spaces are uniquely the most noise stable sets. Furthermore, we prove a quantitative dimension independent versions of uniqueness, showing that a set which is almost optimally noise stable must be close to a half-space. Our work answers a question of Ledoux from 1994 and has numerous applications in theoretical computer science and social choice.
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