  Generalized permutahedra are a family of polytopes with a rich combinatorial
structure and strong connections to optimization. We prove that they are the
universal family of polyhedra with a certain Hopf algebraic structure. Their
antipode is remarkably simple: the antipode of a polytope is the alternating
sum of its faces. Our construction provides a unifying framework to organize
numerous combinatorial structures, including graphs, matroids, posets, set
partitions, linear graphs, hypergraphs, simplicial complexes, building sets,
and simple graphs. We highlight three applications: 1. We obtain uniform proofs
of numerous old and new results about the Hopf algebraic and combinatorial
structures of these families. In particular, we give the optimal formula for
the antipode of graphs, posets, matroids, hypergraphs, and building sets, and
we answer questions of Humpert--Martin and Rota. 2. We show that the
reciprocity theorems of Stanley and Billera--Jia--Reiner on chromatic
polynomials of graphs, order polynomials of posets, and BJR-polynomials of
matroids are instances of the same reciprocity theorem for generalized
permutahedra. 3. We explain why the formulas for the multiplicative and
compositional inverses of power series are governed by the face structure of
permutahedra and associahedra, respectively, answering a question of Loday.
Along the way, we offer a combinatorial user's guide to Hopf monoids.
