Convex Continuations of Some Discrete Functions

We construct convex continuations of discrete functions defined on the vertices of the \( n \)-dimensional unit cube \( [0,1]^n \), an arbitrary cube \( [a,b]^n \), and a parallelepiped \( [c_1,d_1]\times [c_2,d_2]\times \dots \times [c_n,d_n] \). In each of these cases, we constructively prove that, for any discrete function \( f \) defined on the vertices of \( \mathbb {G} \in \{[0,1]^n, [a,b]^n, [c_1,d_1]\times [c_2,d_2]\times \dots \times [c_n,d_n]\} \), first, there exist infinitely many convex continuations to the set \( \mathbb {G} \), and second, there exists a unique function \( f_{DM}\colon \mathbb {G}\to \mathbb {R} \) that is the maximum of convex continuations of \( f \) to \( \mathbb {G} \). We also show that the function \( f_{DM} \) is continuous on \( \mathbb {G} \).