Let $P$ be a convex lattice polytope. Then it has a polynomial Ehrhart function.


I am interested in what can be said about the Ehrhart polynomial when 
$P$ has any of the properties


* is integrally closed
* has a unimodular triangulation
* has a unimodular pulling triangulation
* is compressed


Specifically, does any of these properties imply that the coefficients of the Ehrhart polynomial are non-negative?



Unimodular triangulation and Ehrhart polynomials
I don't think you can generally conclude the nonnegativity of Ehrhart polynomials when expressed in the standard monomial basis; however, the more natural basis for Ehrhart polynomials is given by the binomial coefficients $\binom{n+d} d$, $\binom{n+d-1} d$, ..., $\binom n d$ (here $n$ is the variable of the polynomial and $d$ is the dimension of $P$). Stanley proved in 1980 that the coefficients of the Ehrhart polynomial of $P$ written in terms of this basis are nonnegative. This coefficient vector is known as the $h^\*$-vector, $\delta$-vector, or Ehrhart $h$-vector of $P$. Here are some things that are known and connected to your question:


If $P$ has a unimodular triangulation $T$ then the $h^\*$-vector of $P$ equals the $h$-vector of $T$ (a result contained in the aforementioned 1980 paper by Stanley). (I don't think a pulling triangulation buys anything extra, short of nice properties of the $h$-vector.)


For the case that $P$ is compressed, I believe the state of the art is described in a 2007 paper by Bruns and Roemer in JCTA.


For the case that $P$ is integrally closed, you may consult a recent [preprint](http://arxiv.org/abs/1403.5378) by Braun and Davis.


