Extended generalized permutahedra, and cointeracting bialgebras

A Hopf monoid structure on extended generalized permutahedra (EGP) was recently introduced by M.Aguiar and F.Ardila. We investigate the existence of a cointeracting bialgebra structure on EGP's. We show that a suitable notion of cointeraction exists, not in the classical comodule sense, but via the framework of measuring algebras. The comodule-type map assigns to each polyhedron the sum of pairs of face and tangent cone at the face. EGP's and affine cone EGP's form the cointeracting bimonoids in species with EGP as a third measuring structure. EGP's are in bijection to extended submodular functions. For an EGP, we also describe explicitly the submodular functions of its faces and tangent cones. The braid fan and its relation to preorders play a key role in this description.