The function $f$ maps the positive integer lattice in $\mathbb{R}^n$ (i.e. the vectors in $\mathbb{R}^n$ whose coordinates are all integers) to $\mathbb{R}$. We know that $f$ is convex.
I want to come up with a function $g$ that is a continuation of $f$: that is, $g$ is defined on all of $\mathbb{R}^n_{\ge 0}$, $g$ agrees with $f$ on the integer lattice, and $g$ is convex.
Let $x$ be some positive constant in $\mathbb{R}$. I would also like to be able to find the minimum of $g$ on the box $B = \{v \, | \, 0 \le v_j \le x\}$ in polynomial time.
To summarize, my question:
What technique can I use to create $g$ as a continuation of $f$ such that I can find the minimum of $g$ on $B$ in polynomial time?
By "polynomial time," I mean polynomial time in both $x$ and $n$.
Thanks!