# Counting integral points of a polytope in R^3 (the c_1 coefficient of Ehrhart polynomial)

(Sorry I'm outsider in this field.)

I need to count the number of integral points in a convex polytope in $\mathbf{R}^3$. The cones in the dual fan are not necessarily regular (does it create any problem?)

So, I need the $c_1$ coefficient of the Ehrhart polynomial. There're some formulas (complicated enough for me) but I heard one can express $c_1$ as the sum (over all the edges of the polytope) of the integral lengths of the edges times some correction factors.

1. Can someone give the formula? (In the simple English, please.) Or a reference to something very down-to-earth?
2. In fact I do not need the precise expression but only a very good lower bound. Does something like this exist?

Jamie Pommersheim gave a general formula for $c_1$ in his 1993 paper in Math. Ann. for the case of a tetrahedron. So if you can easily triangulate your polytope, this might be useful.
• yes, thanks. Just I didn't want to enter the subject. I only need this particular coefficient $c_1$ in the particular case of convex polytope in R^3 – Dmitry Kerner Nov 17 '10 at 19:53