  Since their introduction by Stanley~\cite{StanleyOrderPoly} order polytopes
have been intriguing mathematicians as their geometry can be used to examine
(algebraic) properties of finite posets. In this paper, we follow this route to
examine the levelness property of order polytopes. The levelness property was
also introduced by Stanley~\cite{Stanley-CM-complexes} and it generalizes the
Gorenstein property. This property has been recently characterized by
Miyazaki~\cite{Miyazaki} for the case of order polytopes. We provide an
alternative characterization using weighted digraphs. Using this
characterization, we give a new infinite family of level posets and show that
determining levelness is in $\operatorname{co-NP}$. This family can be used to
create infinitely many examples illustrating that the levelness property can
not be characterized by the $h^{\ast}$-vector. We then turn to the more general
family of alcoved polytopes. We give a characterization for levelness of
alcoved polytopes using the Minkowski sum. Then we study several cases when the
product of two polytopes is level. In particular, we provide an example where
the product of two level polytopes is not level.

Combinatorics;Commutative Algebra