## The Annals of Probability

- Ann. Probab.
- Volume 40, Number 3 (2012), 1135-1166.

### Geometric influences

Nathan Keller, Elchanan Mossel, and Arnab Sen

#### 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 ℝ^{n} of Gaussian measure *t*, there exists a coordinate *i* such that the *i*th geometric influence of the set is at least , where *c* is a universal constant. This result is then used to obtain an isoperimetric inequality for the Gaussian measure on ℝ^{n} and the class of sets invariant under transitive permutation group of the coordinates.

#### Article information

**Source**

Ann. Probab., Volume 40, Number 3 (2012), 1135-1166.

**Dates**

First available in Project Euclid: 4 May 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1336136061

**Digital Object Identifier**

doi:10.1214/11-AOP643

**Mathematical Reviews number (MathSciNet)**

MR2962089

**Zentralblatt MATH identifier**

1255.60015

**Subjects**

Primary: 60C05: Combinatorial probability 05D40: Probabilistic methods

**Keywords**

Influences product space Kahn–Kalai–Linial influence bound Gaussian measure isoperimetric inequality

#### Citation

Keller, Nathan; Mossel, Elchanan; Sen, Arnab. Geometric influences. Ann. Probab. 40 (2012), no. 3, 1135--1166. doi:10.1214/11-AOP643. https://projecteuclid.org/euclid.aop/1336136061