We give an interpretation of a class of discrete-to-continuum results for Ising systems
using the theory of zonoids. We define
rational zonotopes and rational zonoids, as the
families of Wulff shapes of perimeters obtained as discrete-to-continuum limits of
finite-range homogeneous Ising systems and of general homogeneous Ising systems,
respectively. Thanks to the characterization of zonoids in terms of measures on the
sphere, rational zonotopes, identified as finite sums of Dirac masses, are dense in the
class of all zonoids. Moreover, we show that a rational zonoid can be obtained from a
coercive Ising system if and only if the corresponding measure satisfies some
connectedness properties, while it is always a continuum limit of
discrete Wulff shapes
under the only condition that the support of the measure spans the whole space.
Finally, we highlight the connection with the homogenization of periodic Ising
systems and propose a generalized definition of rational zonotope of order
,
which coincides with the definition of rational zonotope if
.