Abstract
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 ofsolving an arbitrary system of Boolean equations can be reduced to the problem of minimizing afunction any local minimum of which in the desired region is a global minimum, and thus for thisproblem the problem of local minima is completely resolved.
