Convex Continuation of a Boolean Function and Its Applications

A convex continuation of an arbitrary Boolean function to the set \( [0,1]^n \) is constructed. Moreover, it is proved that for any Boolean function \( f(x_1,x_2,\dots ,x_n) \) that has no neighboring points on the set \( \mathrm{supp} f \), the constructed function \( f_C(x_1,x_2, \dots ,x_n) \) is the only totally maximally convex continuation to \( [0,1]^n \). Based on this, in particular, it is constructively stated that the problem of solving an arbitrary system of Boolean equations can be reduced to the problem of minimizing a function any local minimum of which in the desired region is a global minimum, and thus for this problem the problem of local minima is completely resolved.