Isoperimetric Inequalities Made Simpler

TL;DR

This paper introduces a simple, elementary approach to prove isoperimetric inequalities on the hypercube, providing new proofs and strengthening existing theorems related to function sensitivity and juntas.

Contribution

It offers an elementary proof of classical and recent isoperimetric inequalities and strengthens Friedgut's junta theorem for functions with bounded p-moment sensitivity.

Findings

Elementary proof of Talagrand's isoperimetric inequalities

Proof of a stronger isoperimetric result conjectured by Talagrand

Strengthening of Friedgut's junta theorem for functions with constant p-moment sensitivity

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 $p=1$.