Let $f$ be a balanced Boolean function.
Are there $g$ linear functions, with $$\frac1{2^n}\mathrm{card} \big(\big\{\mathrm{sign} (g (x)) = 2f (x) -1, x \in \{0,1\}^n\big\}\big) > 0.55\quad ?$$
$g (x) = a_1 (2x_1-1) + ... + a_n (2x_n-1)$ and the $a_i$ reals.
Ps : if the answer is yes, then NP=P.