Both volume computation and lattice point enumeration of convex polyhedron are $\#P$ hard. However there is a randomized polytime algorithm for constant factor approximation for volume computation.
Is there a randomized polytime algorithm for constant factor approximation for lattice point enumeration as well?
Is it $\oplus P$ complete to decide if a convex body has odd number of integer points?
If the polytope is convex and also centrally symmetric then what is the situation for 1., 2. and approximate volume computation?