On a Boolean Maximization Problem

Abstract

Among all boolean functions f _j of n variables, n = 2m + 1, the max min (f _j • x _i ) value of correlation between f _j and the n independent binary variables x_i, 1 ≤ i ≤ n, is (^2m _m ).2^-2 m, which is uniquely achieved by f* = maj (x₁,..., x _n ), the majority decision function on the xi’s. This value is asymptotically less, by a factor$\sqrt {\frac{\pi }{2}} \approx 1.2533$of than is achieved by the real vector a* which Achieves$\mathop {\mathop {\max }\limits_{|\alpha | = 1} \mathop {\min }\limits_i }\limits_ (\alpha \cdot {x_i}) = \frac{1}{{\sqrt n }}.$