Convex Influences

TL;DR

This paper introduces convex influence as a new measure for symmetric convex sets in Gaussian space, establishing Gaussian analogues of key Boolean influence theorems and properties, thus extending the analogy between convex sets and Boolean functions.

Contribution

It defines convex influence and proves Gaussian space analogues of fundamental influence theorems for Boolean functions, using novel techniques.

Findings

Gaussian analogues of KKL and sharp threshold theorems

Characterizations of extremal convex sets in Gaussian space

Partial results towards a Gaussian Friedgut's junta theorem

Abstract

We introduce a new notion of influence for symmetric convex sets over Gaussian space, which we term "convex influence". We show that this new notion of influence shares many of the familiar properties of influences of variables for monotone Boolean functions $f: \{\pm1\}^n \to \{\pm1\}.$ Our main results for convex influences give Gaussian space analogues of many important results on influences for monotone Boolean functions. These include (robust) characterizations of extremal functions, the Poincar\'e inequality, the Kahn-Kalai-Linial theorem, a sharp threshold theorem of Kalai, a stability version of the Kruskal-Katona theorem due to O'Donnell and Wimmer, and some partial results towards a Gaussian space analogue of Friedgut's junta theorem. The proofs of our results for convex influences use very different techniques than the analogous proofs for Boolean influences over $\{\pm1\}^n$. Taken as a whole, our results extend the emerging analogy between symmetric convex sets in Gaussian space and monotone Boolean functions from $\{\pm1\}^n$ to $\{\pm1\}$