Is a finite integer subset of a convex real set convex?

Question

Specifically, can I take a convex real set, show that the definition of convexity holds for it, and then make claims based on that definition of convexity for an integer subset? I know that the interior of a convex set is convex; how many of those properties can I bring with me to an integer subset of the same?

Answer 1

It seems the following.

Some properties of a convex set $A\subset\mathbb R^n$ are inherited by its integer subset $A_I=A\cap \mathbb Z^n$. For instance, if $x,y\in A_I$ and a point $z\in \mathbb Z^n$ lies on the segment between $x$ and $y$ then $z\in \mathbb Z^n\cap A=A_I$. If $z\in \mathbb Z^n\setminus A_I$ then $z\not\in A$ so the $z$ can be separated from $A$ (and, hence, from $A_I$) by a hyperplane.