Let $K$,$L$,$M$ be convex lattice polytopes (so their vertices are in $\mathbb{Z}^n$) in $\mathbb{R}^n$ satisfying $K+L\subseteq M+M$ (Minkowski sum). Do we always have
$$|K\cap\mathbb{Z}^n|\cdot|L\cap\mathbb{Z}^n|\le |M\cap\mathbb{Z}^n|^2 ?$$


I looked at books/papers on Erhart polynomial and Brunn-Minkowski inequality etc., but did not find an satisfying answer. Any comment? Thanks.



Inequality of arithmetic and geometric means for the lattice polytopes?
This inequality does not necessarily hold, at least for $n\geq 3$. It is somehow connected with the fact that there is no Pick's formula in more than two dimensions since there exists a convex lattice polytope with a large volume but containing a small number of lattice points. 


So, for instance, for $n=3$ let $M$ be the convex hull of the points $(0,0,0)$, $(1,1,0)$, $(0,1,2k)$ and $(1,0,2k)$. Then $|M\cap {\mathbb Z}^3|=4$, but $(M+M)\cap {\mathbb Z}^3\supset \{(1,1,t):0\leq t\leq 2k\}$. So $K$ and $L$ can be chosen as vertical segments containing $k$ lattice points each.


