Concentration and influences

Abstract

Consider the discrete cube Ω={0,1}^N, provided with the uniform probabilityP. We denote byd(x, A) the Hamming distance of a pointx of Ω and a subsetA of Ω. We define the influenceI(A) of theith coordinate onA as follows. Forx in Ω, consider the pointT _i (x) obtained by changing the value of theith coordinate. Then$I_i (A) = P(\{ x \in A;T_i (x) \notin A\} ).$

We prove that we always have$P(A)\int_\Omega {d(x,A)dP(x) \leqslant \frac{1}{2}\sum\limits_{i \leqslant N} {I_i (A).} }$

Since it is easy to see that$\sum\nolimits_{i \leqslant N} {I_i (A)^2 \leqslant \frac{1}{4}}$, this recovers the well known fact that ∫_Ω d(x, A)dP(x) is at most of order$\sqrt N$ whenP(A)≥1/2. The new information is that ∫_Ω d(x, A)dP(x) can be of order$\sqrt N$ only ifA reassembles the Hamming ball {x; ∑_1≤N x _i ≥N/2}.