Given are two convex bodies $K, L \subset \mathbb{R}^n$ that contain the origin as an interior point. Assume the number of integer points contained in $\lambda K$ equals the number of integer points contained in $\lambda L$ *for every $\lambda > 0$.* Is $K$ equal to $L$ modulo unimodular transformations?

If not, in what way is $K$ similar to $L$ besides the well-known fact that they must have the same volume?

**Addendum:** Anton's example shows that $K$ and $L$ are not necessarily equal modulo unimodular transformations even when they are taken to be two integral polygons in the plane and Abhinav Kumar points out in the comments that in the case $K$ is a integral polytope the number of integer points inside $n K$ ($n$ a natural number) is the value of the Ehrhart polynomial of $K$ at $n$.

**Added 21/11/2013:** For $K$ and $L$ ellipsoids this problem translates to "what can we say about isospectral flat Riemannian tori?". Therefore, by a famous remark by Milnor there exist ellipsoids $K$ and $L$ in $\mathbb{R}^{16}$ that are not unimodularly equivalent and for which the number of integer points contained in $\lambda K$ equals the number of integer points contained in $\lambda L$ for every $\lambda > 0$.