Consider the bounded (closed) convex region given by:
$a^Tx = b$ , $a^Tx = b + ||a||_2$ and $ m_1 \le x_i \le M_2 $ forall $1 \le i < n$ and $m_2 \le x_n \le M_2$ i.e two parallel hyperplanes at at unit perpedicular distance from each other within the $\mathbb{R}^n$ equivalent version of a `cuboid'.
Can we find an upper bound to the number of lattice points (points in $\mathbb{Z}^n$) contained in this region - is there some theory regarding this in the literature?
My Idea My intuition suggested that the number of lattice points should be bounded by the volume. However I couldn't show this formally, moreover I am not sure what the volume of this region would be either?