Given a convex polytope let the width of the polytope be $d$ and the farthest euclidean distance between any points in the polytope be $e$.
Denote $\mathcal P(a,c)$ to be the set of convex polytopes in $\Bbb R^n$ presented by $O(n^a)$ linear inequalities with $e/d=O(n^c)$.
Is it $\#P$ complete to find number of integer points in such polytopes?
Is it $NP$ complete to decide existence of integer points in such polytopes?
I am essentially asking if the polytope is reasonably round what is the difficulty of counting and decision problems. Here for quantifying roundedness I use width and max distance (but may be some other quantification which I do not know may be appropriate).
I think some strange partition of 0/1 cube with convex faces might suffice for 1.