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A new definition of influences of Boolean functions

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Discrete Analysis

The notion of influences of variables on Boolean functions is one of the central concepts in the theory of discrete harmonic analysis. 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. 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. I will also discuss some statistical connection to this problem. This is joint work with Nathan Keller and Elchanan Mossel

This talk is part of the Isaac Newton Institute Seminar Series series.

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