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. Bibliogr. 15.