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arXiv · 2204.06686

Isoperimetric Inequalities Made Simpler

Ronen Eldan·Guy Kindler·Noam Lifshitz·Dor Minzer
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Abstract

We give an alternative, simple method to prove isoperimetric inequalities over the hypercube. In particular, we show: 1. An elementary proof of classical isoperimetric inequalities of Talagrand, as well as a stronger isoperimetric result conjectured by Talagrand and recently proved by Eldan and Gross. 2. A strengthening of the Friedgut junta theorem, asserting that if the $p$-moment of the sensitivity of a function is constant for some $1/2 + \varepsilon\leq p\leq 1$, then the function is close to a junta. In this language, Friedgut's theorem is the special case that $p=1$.

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Ronen Eldan, Guy Kindler, Noam Lifshitz, Dor Minzer. 2022-04-14. Isoperimetric Inequalities Made Simpler. https://arxiv.org/abs/2204.06686

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