  Continuing a well established tradition of associating convex bodies to
monomial ideals, we initiate a program to construct asymptotic Newton polyhedra
from decompositions of monomial ideals. This is achieved by forming a graded
family of ideals based on a given decomposition. We term these graded families
powers since they generalize the notions of ordinary and symbolic powers.
Asymptotic invariants for these graded families are expressed as solutions to
linear optimization problems on the respective convex bodies. This allows to
establish a lower bound on the Waldschmidt constant of a monomial ideal by
means of a more easily computable invariant, which we introduce under the name
of naive Waldschmidt constant.
