On the Eldan-Gross inequality

TL;DR

This paper provides an alternative proof of the Eldan-Gross inequality, extending its applicability to biased hypercubes and spaces with positive Ricci curvature bounds, enhancing understanding of Boolean function sensitivity.

Contribution

It offers a new proof of the Eldan-Gross inequality that applies to biased hypercubes and spaces with positive Ricci curvature, broadening its scope.

Findings

The inequality holds for biased hypercubes.

The proof extends to spaces with positive Ricci curvature.

The result enhances understanding of Boolean function sensitivity.

Abstract

A recent discovery of Eldan and Gross states that there exists a universal $C>0$ such that for all Boolean functions $f:\{-1,1\}^n\to \{-1,1\}$, $$ \int_{\{-1,1\}^n}\sqrt{s_f(x)}d\mu(x) \ge C\text{Var}(f)\sqrt{\log \left(1+\frac{1}{\sum_{j=1}^{n}\text{Inf}_j(f)^2}\right)} $$ where $s_f(x)$ is the sensitivity of $f$ at $x$, $\text{Var}(f)$ is the variance of $f$, $\text{Inf}_j(f)$ is the influence of $f$ along the $j$-th variable, and $\mu$ is the uniform probability measure. In this note, we give an alternative proof that applies to biased discrete hypercube, and spaces having positive Ricci curvature lower bounds in the sense of Bakry and \'Emery.