  Generalized permutohedra are deformations of regular permutohedra, and arise
in many different fields of mathematics. One important characterization of
generalized permutohedra is the Submodular Theorem, which is related to the
deformation cone of the Braid fan. We lay out general techniques for
determining deformation cones of a fixed polytope and apply it to the Braid fan
to obtain a natural combinatorial proof for the Submodular Theorem.
  We also consider a refinement of the Braid fan, called the nested Braid fan,
and construct usual (respectively, generalized) nested permutohedra which have
the nested Braid fan as (respectively, refining) their normal fan. We extend
many results on generalized permutohedra to this new family of polytopes,
including a one-to-one correspondence between faces of nested permutohedra and
chains in ordered partition posets, and a theorem analogous to the Submodular
Theorem. Finally, we show that the nested Braid fan is the barycentric
subdivision of the Braid fan, which gives another way to construct this new
combinatorial object.
