For integral polytopes, it is conjectured (T. Hibi), that if the [$h^\*$-vector](http://en.wikipedia.org/wiki/Ehrhart_polynomial#Ehrhart_Series) is symmetric, then it is also unimodal (increasing, then non-decreasing).


A non-integral polytope do not, in general, have a polynomial [Ehrhart function](http://en.wikipedia.org/wiki/Ehrhart_polynomial).
However, if it does, we can compute the corresponding $h^\*$-vector.


Now, are there explicit examples where the conjecture mentioned above is false,
published somewhere? That is, that the conjecture does not extend to non-integral polytopes of this special kind above?


I stumbled upon such an example in my current research, and I am just wondering if it is worth mentioning. The polytope in question is the Gelfand-Tsetlin polytope,
$GT(\lambda,w)$ for $\lambda=(2,2,2)$ and $w=(1,1,1,1,1,1)$.



On a conjecture by Hibi regarding h-vectors
The conjecture is false for integral polytopes in dimension $\ge 6$; see Mircea Mustata and Sam Payne, Ehrhart polynomials and stringy Betti numbers, Math. Ann. 333 (2005), no. 4, 787-795 ([arXiv version](http://arxiv.org/abs/math/0504486)), and Sam Payne, Ehrhart series and lattice triangulations, Discrete Comput. Geom. 40 (2007), 365-376 ([arXiv version](http://arxiv.org/abs/math/0702052)). For a rational version of Hibi's palindromic theorem, you might want to check out [this paper by Matthew Fiset and Alexander Kasprzyk](http://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1n18). The unimodality conjecture for *integrally closed* reflexive polytope is still open; see, e.g., [this recent paper by Ben Braun and Robert Davis](http://arxiv.org/abs/1403.5378).


