(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.

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