Geometric influences

TL;DR

This paper introduces a geometric definition of influences in continuous product spaces, establishes bounds analogous to classical influence inequalities, and applies these results to isoperimetric problems and symmetric sets in Gaussian space.

Contribution

It proposes a new geometric influence measure for continuous spaces and proves bounds similar to KKL and Talagrand's inequalities, extending influence theory to Gaussian and symmetric settings.

Findings

Established a tight KKL-type bound for Gaussian measures.

Proved influence sum bounds analogous to Talagrand's inequalities.

Derived an isoperimetric inequality for Gaussian measures and symmetric sets.

Abstract

We present a new definition of influences in product spaces of continuous distributions. Our definition is geometric, and for monotone sets it is identical with the measure of the boundary with respect to uniform enlargement. We prove analogs of the Kahn-Kalai-Linial (KKL) and Talagrand's influence sum bounds for the new definition. We further prove an analog of a result of Friedgut showing that sets with small "influence sum" are essentially determined by a small number of coordinates. In particular, we establish the following tight analog of the KKL bound: for any set in $\mathbb{R}^n$ of Gaussian measure $t$, there exists a coordinate $i$ such that the $i$th geometric influence of the set is at least $ct(1-t)\sqrt{\log n}/n$, where $c$ is a universal constant. This result is then used to obtain an isoperimetric inequality for the Gaussian measure on $\mathbb{R}^n$ and the class of sets invariant under transitive permutation group of the coordinates.