Concave continuations of Boolean-like functions and some of their properties

In this paper, we present concave continuations of discrete functions defined on the vertices of an $n$-dimensional unit cube, an $n$-dimensional arbitrary cube, and an $n$-dimensional arbitrary parallelepiped. It is constructively proved that, firstly, for any discrete function $f_D$ defined on the vertices of $\mathbb{G}$, where $\mathbb{G}$ is one of these three sets, the cardinality of the set of its concave continuations on $\mathbb{G}$ is equal to infinity, and, secondly, there is a function $f_{NR}$ that is the minimum among all its concave continuations on $\mathbb{G}$. The uniqueness and continuity of the function $f_{NR}$ on $\mathbb{G}$ are also proved.