Decision problems in Integer Linear Programming have Lenstra type algorithms (https://www.math.leidenuniv.nl/~lenstrahw/PUBLICATIONS/1983i/art.pdf) have been generalized to convex integer program techniques (See Complexity of integer quasiconvex polynomial optimization by Sebastian Heinz).

What is the difficulty in generalizing Barvinok type counting integer points in Linear Integer Programming techniques (https://barvinok.sourceforge.io/) to counting integer points in convex integer programming?