Question

Conjecture (Ehrhart). If a convex body $K \subset {\mathbb R}^n$ has its barycenter at the origin and contains no other point with integer coordinates, the volume of $K$ is less than or equal to $(n + 1)^n/n!$.

Ehrhart proved this for $n = 2$ and for simplices in any dimension (see his paper in J. Reine Angew. Math. 305, (1979) 218-220 and the references therein). These results and the conjecture are also cited in the book "Unsolved Problems in Geometry" by Croft, Falconer, and Guy.

Does anybody know whether anything interesting has been done on this conjecture?

For example, is the following weaker version of the conjecture known to be true?

Weaker version of Ehrhart's conjecture. There exists a universal constant $C > 1$ such that the volume of every convex body $K \subset {\mathbb R}^n$ satisfying the hypotheses of the conjecture is less than or equal to $C^n (n + 1)^n/n!$.

Answer 1

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.