Given a set $S$ of $\{0, 1\}^{n}$ points - that is, $n$-length points where each entry is one of $0$ or $1$ - I am interested in the lattice points of its convex hull, $conv(S)$. It seems pretty obvious to me that the set of lattice points of $conv(S)$ is exactly $S$ itself, but I don't quite know how to go about proving it.
For clarification, by "lattice point", I mean a point where every entry is an integer.
Suppose that $v$ is a lattice point, $v\notin S$, and $v\in conv(S)$. Then $v$ is a convex combination of elements of $S$. Suppose that for some $k$ the $k$th component of $v$ is $0$. Then the $k$th component of each element of $S$ must be $0$, so the $0$ components gives no information.
We need only consider the case where each component of $v$ is $1$. But this is an extreme point of $[0,1]^n$ and not in the convex hull of $[0,1]^n\setminus\{v\}$, let alone the set lattice points in $[0,1]^n\setminus\{v\}$.