Richard Stanley [showed](http://math.mit.edu/~rstan/pubs/pubfiles/66.pdf) that order polytopes have a unimoudlar triangulation.
In particular, this implies that they are integrally closed/normal. 


One can generalize order polytopes to marked order polytopes,
where the smallest and largest elements are not always 0 and 1 respectively,
but other integers, and the poset do not have unique top and bottom.


For example, Gelfand-Tsetlin polytopes can be realized as the construction above, and it is quite easy to show that these particular polytopes are integrally closed. 


*However, is it known if ALL marked order polytopes are integrally closed?*


*Reminder:* A convex polytope $P$ with integer vertices is integrally closed if
we have that each integer point $p$ in $kP$ can be expressed as
$p=p\_1+\dots+p\_k$ where each $p\_i$ is an integer point in $P$.



Are all marked order polytopes normal?
I have found a positive answer to this question: One can interpret $p$
as a point in some $kP'$ where $P'$ is the poset where the unknowns are totally ordered, and have fixed lower and upper bounds. 
This poset $P'$ can then be seen as a face in a GT-polytope,
so using that the GT-polytopes are integrally closed, it follows that $p$ 
can be expressed as a sum of points in $P'$, and the inequalities between coordinates are by construction compatible with those in $P$.


Actually, one can show a stronger statement, that each such polytope has regular, unimodular triangulation.


