We know that both volume computation and lattice point enumeration of convex polyhedron is $\#P$ hard. However there is a randomized polytime algorithm for constant factor approximation for volume computation.

1. Is there a randomized polytime algorithm for constant factor approximation for lattice point enumeration as well?

2. 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?

3. Given two polytopes is it in $PP$ to decide which has larger volume and is it in $PP$ to decide which has more lattice points?