Stability of Talagrand's influence inequality

09/26/2019

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by   Ronen Eldan, et al.

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We strengthen several classical inequalities concerning the influences of a Boolean function, showing that near-maximizers must have large vertex boundaries. An inequality due to Talagrand states that for a Boolean function f, var(f)≤ C∑_i=1^nInf_i(f)/1+log(1/Inf_i(f)), where Inf_i(f) denotes the influence of the i-th coordinate. We give a lower bound for the size of the vertex boundary of functions saturating this inequality. As a corollary, we show that for sets that satisfy the edge-isoperimetric inequality or the Kahn-Kalai-Linial inequality up to a constant, a constant proportion of the mass is in the inner vertex boundary. Our proofs rely on new techniques, based on stochastic calculus, and bypass the use of hypercontractivity common to previous proofs.