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 C, where C is one of these three sets, the cardinality of the set of its concave continuations on C is equal to infinity, and, secondly, there is a function f(NR) that is the minimum among all its concave continuations on C. The uniqueness and continuity of the function f(NR) on C are also proved.