Ehrhart polynomials

Question

I was reading Richard Stanley's article Two Poset Polytopes, and when he defined the Ehrhart polynomial of a convex polytope $P$, he mentioned that if $P$ is an $n$- dimensional polytope in $\mathbb{R}^n$, then the leading coefficient of the Ehrhart polynomial is the volume of $P$. Is this volume the usual notion of volume or it represents something else? Thanks!

Answer 1

Yes, it's the usual volume, at least when the lattice is $\Bbb Z^n$.

If we take the union of unit cubes centred at points of $m P\cap\Bbb Z^n$ and then shrink by a factor of $m$ you get a set of volume $m^{-n}|P\cal\Bbb Z^n|$ which approximates $P$ more closely as $m\to\infty$. Therefore $|P\cap{\Bbb Z}^n|\sim m^n\text{Vol}(P)$.