Geometric Influences

Details Geometric Influences Geometric Influences

2009-11-09

arxiv.org/abs/0911.1601v3

Mathematics

Probability

minor changes, a brief statistical application added

Scientific paper

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 analogues of the Kahn-Kalai-Linial (KKL) and Talagrand's influence sum bounds for the new definition. We further prove an analogue 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 analogue 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.

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