  The Brion morphism maps a generalized permutahedron to a collection of posets
associated to its vertices. We compute this map explicitly for the Hopf monoids
of permutahedra, associahedra, and orbit polytopes, and we explore the dual
Brion map of the primitive Lie monoids associated to these three Hopf monoids.
We describe the Lie monoid structure of the primitives in this dual setting and
in particular we show that the Lie monoid of primitives of associahedra is
isomorphic to the positive part of the Witt Lie algebra
