  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 $(n+1)$-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.
  We construct a subdivision of any lifted generalized permutahedron whose
pieces are indexed by compositions. The volume of each piece is given by a
polynomial whose combinatorial properties we investigate. We show how this
"composition polynomial" arises naturally in the polynomial interpolation of an
exponential function. We prove that its coefficients are positive integers, and
present evidence suggesting that they may also be unimodal.
