Direct and inverse mean value properties for polylinear functions and their application

This paper is devoted to the formulation and proof of the theorems on the mean value of a polylinear function, similar to the direct and inverse theorems on the mean value of harmonic functions. It is proved that the value of an arbitrary polylinear function $f_P(x)$ at the central point of $\mathbb{G}$—an arbitrary $n$-dimensional coordinate parallelepiped—is equal to the mean value of the function $f_P(x)$ over the set of $k$-dimensional faces $\mathbb{G}$ for any $k\in\{0,\ldots,n\}$. Based on this, it is justified that just once, by calculating the value of the polylinear continuation $f_P(x)$ of an arbitrary Boolean function $f_B(x)$ at the central point of an $n$-dimensional unit cube, one can find the number of Boolean vectors on which the Boolean function $f_B(x)$ takes the value 1 and thereby, in particular, determine the satisfiability of the Boolean function $f_B(x)$. It has also been established that such a property is characteristic only of polylinear functions, i.e., it has been proven that if for any $\mathbb{G}$ — $n$-dimensional coordinate parallelepiped and at least for some number $k\in\{0,\ldots,n\}$, the value of the continuous function $f(x)$ at the central point $\mathbb{G}$ is equal to the mean value of the function $f(x)$ over the set of $k$-dimensional faces of $\mathbb{G}$, then the function $f(x)$ is polylinear.