  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. We
introduce a novel family of irreducible powers.
  Irreducible powers and symbolic powers of monomial ideals are studied by
means of the corresponding irreducible polyhedron and symbolic polyhedron
respectively. 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, an asymptotic invariant which can be defined using the symbolic
polyhedron, by means of an analogous invariant stemming from the irreducible
polyhedron, which we introduce under the name of naive Waldschmidt constant.
