Lattice polytope

Remarks

Example: the triangle with vertices $(0,0)$, $(3,0)$, $(0,2)$ is a lattice polygon, since all three vertices have integer coordinates.

Non-example: the triangle with vertices $(0,0)$, $(1,0)$, $(0, 1/2)$ is not a lattice polygon, since the vertex $(0, 1/2)$ is not in $\Z^2$. It is a rational polytope: its vertices lie in $\Q^2$.

More generally, for a lattice $\Lambda \subseteq \R^n$, a lattice polytope with respect to $\Lambda$ is a convex polytope with all vertices in $\Lambda$. A linear change of coordinates turns $\Lambda$ into $\Z^n$, so the case $\Lambda = \Z^n$ is the standard one. Two lattice polytopes are usually regarded as the same when an affine map $x \mapsto Ax + b$ with $A \in GL_n(\Z)$ and $b \in \Z^n$ carries one onto the other.