Work

On the Eldan–Gross inequality

Year: 2025

Type: article

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 → { − 1 , 1 } , ∫ { − 1 , 1 } n s f ( x ) d μ ( x ) ≥ C Var ( f ) log ⁡ ( 1 + 1 ∑ j = 1 n Inf j ( f ) 2 ) where s f ( x ) is the sensitivity of f at x , Var ( f ) is the variance of f , 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.

Source: Journal of Functional Analysis

DOI: 10.1016/j.jfa.2025.111255

Authors: Paata Ivanisvili, Haonan Zhang

Institutions: University of California, Irvine, University of South Carolina

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,250

Funder: Division of Mathematical Sciences