Isoperimetric Inequalities Made Simpler

04/14/2022

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by   Ronen Eldan, et al.

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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 p≥ 1/2 + ε, then the function is close to a junta. In this language, Friedgut's theorem is the special case that p=1.