HALF-SPACES WITH INFLUENTIAL VARIABLE

We consider Boolean functions f defined on Boolean cube {-1, 1}(n) of half-spaces, i.e., functions of the form f (x) = sign(omega.x-theta). Half-space functions are often called linear threshold functions. We assume that the Boolean cube {-1,1}(n) is equipped with a uniform measure. We also assume that parallel to omega parallel to(2 )<= 1 and parallel to omega parallel to(infinity) = delta for some delta > 0. Let 0 <= epsilon <= 1 be such that vertical bar Ef vertical bar <= 1 - epsilon. We prove that there exists a constant C > 0 such that max,(Inf(i),f) >= C delta epsilon, where Inf(i) f denotes the influence of the ith coordinate of the function f. This establishes the lower bound obtained earlier by Matulef et al. [SIAM J. Comput., 39 (2010), pp. 2004-2047]. We also show that the optimal constant in this inequality exceeds 3-root 2/64 approximate to 0.066. As an auxiliary result we prove a lower bound for small ball inequalities of linear combinations of Rademacher random variables.