Isoperimetric Inequalities Made Simpler

2022-04-14

arXiv.org

AI Summary

This paper addresses the proof complexity of isoperimetric inequalities and junta theorems on the hypercube. Methodologically, it introduces a unified, elementary combinatorial framework integrating Boolean function sensitivity theory, discrete isoperimetric estimates, and $p$-th moment analysis. It establishes, for the first time, a quantitative connection between $p$-biased sensitivity and junta structure for $p in [1/2+varepsilon, 1]$. Key contributions are: (1) a concise, elementary proof of Talagrand’s classical isoperimetric inequality; (2) confirmation and strengthening of his conjecture, yielding an Eldan–Gross-type improved isoperimetric inequality; and (3) a generalization of Friedgut’s junta theorem to $p in [1/2+varepsilon, 1]$, achieving $O(1)$-junta approximation under constant $p$-biased sensitivity—significantly improving upon the prior restriction to $p = 1$.

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

Problem

• Proves classical isoperimetric inequalities simply
• Strengthens Friedgut junta theorem for sensitivity
• Simplifies Talagrand's conjectured isoperimetric results

Innovation

• Simpler proof method for hypercube isoperimetric inequalities
• Strengthened Friedgut junta theorem with p-moment condition
• Elementary proof of Talagrand's classical inequalities