Lifted generalized permutahedra and composition polynomials

Generalized permutahedra are the polytopes obtained from the permutahedron by changing the edge lengths while preserving the edge directions, possibly identifying vertices along the way. We introduce a “lifting” construction for these polytopes, which turns an n  -dimensional generalized permutahedron into an (n+1)$(n+1)$-dimensional one. We prove that this construction gives rise to Stasheff ?s multiplihedron from homotopy theory, and to the more general “nestomultiplihedra”, answering two questions of Devadoss and Forcey.