Isoperimetric Inequalities Made Simpler

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 $p=1$.