Let's say I have a bunch of linear functions $f_1,\cdots,f_n$ in $k$ variables; then $f_1,\cdots, f_n\le0$ defines a polyhedron $P$ in the $k$-dimensional space.
What I'm looking for is going the other way: given data points inside a polyhedron $P$, is there an efficient/fast way to find the functions $f_1,\cdots,f_n$ cutting the polyhedron? Those data points are finitely many integer points (i.e. points whose all coordinates are integers) and $P$ is "the smallest" polyhedron containing them. Basically I require two things:
$P$ does not contain any other integer points.
The linear functions themselves have integer coefficients.
A naive way would be to loop over all possible candidate functions (their integer coefficients) in a certain range but there should be a much better/faster/efficient way of doing it.
I am not specifically looking for an answer to this problem, rather a reference or direction for where I should be looking at. Has this been studied at all? I understand that this is not exactly what linear/integer programming is about, so apologies if the question is not appropriate for this site.