Abstract
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 \(\mathbb {G}\) .
