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 $\mathbb{G}$. Bibliogr. 24.