  Nestohedra are a family of convex polytopes that includes permutohedra,
associahedra, and graph associahedra. In this paper, we study an extension of
such polytopes, called extended nestohedra. We show that these objects are
indeed the boundaries of simple polytopes, answering a question of Lam and
Pylyavskyy. We also study the duals of (extended) nestohedra, giving a complete
characterization of isomorphisms (as simplicial complexes) between the duals of
extended nestohedra and a partial characterization of isomorphisms between the
duals of nestohedra and extended nestohedra. In addition, we give formulas for
their $f$-, $h$-, and $\gamma$-vectors. This includes showing that the
$f$-vectors of the extended nestohedron corresponding to a forest $F$ and the
nestohedron corresponding to the line graph of $F$ are the same, as well as
showing that all flag extended nestohedra have nonnegative $\gamma$-vectors,
thus proving Gal's conjecture for a large class of flag simple polytopes. We
also relate the $f$- and $h$-vectors of the nestohedra and extended nestohedra,
as well as give explicit formulas for the $h$- and $\gamma$-vectors in terms of
descent statistics for a certain class of flag extended nestohedra. Finally, we
define a partial ordering on partial permutations that is a join semilattice
quotient of the weak Bruhar order on the symmetric group, and such that any
linear extension of the partial order provides a shelling of the dual of the
stellohedron.
