I guess it must be in bad taste to answer my own question, but I now know a bit more about this problem and maybe there are other people interested in references to it.

As I mentioned in the question, Ehrhart himself settled the $2$-dimensional case and the case where the convex body is a simplex. Apparently nothing interesting was done until late last year when Robert J. Berman and Bo Berndtsson http://arxiv.org/abs/1112.4445 used PDE techniques to settle the case where the convex body is *dual to a Fano polytope*.

**Reminder.** A Fano polytope is a lattice polytope such that the vertices of each facet define a basis of the lattice.

As for the weak version of the Ehrhart conjecture: it is true. The preprint will be soon on the ArXiv.

**Addendum.** Here is the proof of the weak (or asymptotic) version of the Ehrhart conjecture modulo the following result that will appear in a forthcoming paper by Balacheff, Tzanev, and myself.

**Theorem (ABT).** If the origin is the unique integer point in the interior of a convex body $K \subset \mathbb{R}^n$, then the volume of the dual body $K^*$ is at least $(\pi/8)^n (n+1)/n!$.

The conjecture is that the volume of $K^*$ is at least $(n+1)/n!$, but we were able to show this only for $n=2$.

Now for the weak version of the Ehrhart conjecture: let $K \subset \mathbb{R}^n$ be a convex body with its barycenter at the origin and containing no other integer point. By the Blaschke-Santalo inequality (actually a version of it that holds for asymmetric bodies and their duals with respect to the barycenter, but I don't remember to whom it is due!!), we have
that $\epsilon_n^2 \geq |K||K^*|$, where $\epsilon_n$ is the volume of the Euclidean unit ball.

This inequality, together with the ABT theorem yields
$$
\epsilon_n^2 \geq |K||K^*| \geq |K| (\pi/8)^n (n+1)/n!
$$
or $|K| \leq (8/\pi)^n\epsilon_n^2 n!/(n+1)$.

Now, the proof reduces to showing that the quantity
$$
\left(\epsilon_n^2 (n!)^2/(n+1)^{(n+1)}\right)^{1/n}
$$
is bounded above by a quantity $C$. This is just the $n$th root of the quotient of the volume product of the ball and the volume product of the simplex and, therefore, standard fare in asymptotic geometry. This quantity is asymptotically $2\pi / e$ and *seems* to be always less than $3$. I'll check and edit when I have a bit more time.

I worked out this proof at the early stages of my collaboration with Balacheff (before Tzanev joined us), but this result got dropped out of the (forthcoming) paper. I reproduce it here from my notes, but I'm a bit rusty on the details. Very possibly, one can bypass ABT and apply a judicious mixture of Minkowki's lattice point theorem, Rogers-Shephard inequality, the Bourgain-Milman theorem, and the asymmetric version of Blaschke_Santalo to get the proof. In any case, those are the basic ingredients that go in the proof above.