On the Eldan-Gross inequality

TL;DR

The note provides an alternative, broadly applicable proof of the Eldan–Gross-type inequality that originally held for the Boolean cube, showing it extends to biased hypercubes and to spaces with a positive Bakry–Émery curvature (CD(K,∞)). The authors leverage Bobkov–Götze-type isoperimetric inequalities and hypercontractivity on biased cubes, and then generalize the framework to continuous Markov triples, where diffusion and curvature enable analogous bounds. They derive explicit lower bounds on the expected gradient norm in terms of the variance and either the total squared influences or a variance-based logarithmic term, thereby unifying discrete and continuous settings under a common isoperimetric-diffusion approach. This broadens the applicability of Eldan–Gross-type concentration phenomena and links to Talagrand-type inequalities across a spectrum of spaces.

Abstract

A recent discovery of Eldan and Gross states that there exists a universal $C>0$ such that for all Boolean functions $f:\{-1,1\}^n\to \{-1,1\}$, $$ \int_{\{-1,1\}^n}\sqrt{s_f(x)}dμ(x) \ge C\text{Var}(f)\sqrt{\log \left(1+\frac{1}{\sum_{j=1}^{n}\text{Inf}_j(f)^2}\right)} $$ 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 $μ$ 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 Émery.

Theorems & Definitions (13)

• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Proposition 5
• Lemma 6
• proof
• Proposition 7
• proof
• proof : Proof of Theorem \ref{['thm:eg biased']}