Does there exists a universal constant $C \geq 1$ such that if $X$ is any a smooth, toric, Fano $n$-dimensional manifold admitting a Kähler-Einstein metric, then its anticanonical degree $(-K\_X)^n$ is at most $C^n (n+1)^n$?


This would follow from the weakening of Ehrhart's conjecture that I proposed in [Reference request: Ehrhart's conjecture on the geometry of numbers](https://mathoverflow.net/questions/88153/reference-request-ehrharts-conjecture-on-the-geometry-of-numbers) (or at least this is what I understand from reading page 6 of the paper of Nill and Paffenholtz <https://arxiv.org/abs/0905.2054>).


I wonder:


**1.** Is this known?


**2.** Is this interesting?


My knowledge of algebraic geometry is pitiful so please do not be offended if the question is really naive. I'm just trying to see what possible interesting consequences the weakened conjecture could have.



Bound on the (anticanonical) degree of toric Fano varieties
I think the answer to this is no. In fact according to [1] there is no universal polynomial bound on the $n$-th root of $c\_1(X)^n$ as X runs over all toric Fanos of dimension n (this is referenced to Debarre but I am afraid I do not have this source with me at the moment).


By contrast the purpose of [1] is to prove that such a bound does exist if you restrict to Kahler-Einstein metrics. 


[1] The projective space has maximal volume among all toric Kähler-Einstein manifolds
Robert J. Berman, Bo Berndtsson
<http://arxiv.org/abs/1112.4445>


