Theory of Probability &Its Applications, Volume 65, Issue 1, Page 114-120, January 2020.
We consider Boolean functions $f$ defined on Boolean cube $\{-1,1\}^n$ of half-spaces, i.e., functions of the form $f(x)=\operatorname{sign}(\omega\cdot x-\theta)$. 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 $\|\omega\|_2\leq 1$ and $\|\omega\|_{\infty} = \delta$ for some $\delta>0$. Let $0\leq\varepsilon\leq 1$ be such that $|\mathbf{E} f|\leq 1-\varepsilon$. We prove that there exists a constant $C>0$ such that $\max_i(\operatorname{Inf}_i f)\geq C\delta\varepsilon$, where $\operatorname{Inf}_i f$ denotes the influence of the $i$th 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 $3\sqrt{2}/64\approx 0.066$. As an auxiliary result we prove a lower bound for small ball inequalities of linear combinations of Rademacher random variables.

中文翻译：

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具有影响变量的半空间

概率论及其应用，第65卷，第1期，第114-120页，2020年1月。
我们考虑在半角的布尔立方体$ \ {-1,1 \} ^ n $上定义的布尔函数$ f $，即形式为$ f（x）= \ operatorname {sign}（\ omega \ cdot x- \ theta）$。半空间函数通常称为线性阈值函数。我们假设布尔立方体$ \ {-1,1 \} ^ n $配备了统一的度量。我们还假设$ \ | \ omega \ | _2 \ leq 1 $和$ \ | \ omega \ | _ {\ infty} = \ delta $对于某些$ \ delta> 0 $。令$ 0 \ leq \ varepsilon \ leq 1 $等于$ | \ mathbf {E} f | \ leq 1- \ varepsilon $。我们证明存在一个常数$ C> 0 $，使得$ \ max_i（\ operatorname {Inf} _i f）\ geq C \ delta \ varepsilon $，其中$ \ operatorname {Inf} _i f $表示函数$ f $的$ i $ th坐标。这确定了Matulef等人早先获得的下限。[SIAM J.Comput。，39（2010），第2004--2047页]。我们还表明，该不等式中的最佳常数超过$ 3 \ sqrt {2} / 64 \约0.066 $。作为辅助结果，我们证明了Rademacher随机变量线性组合的小球不等式的下限。