  We study multivariate Mellin transforms of Laurent polynomials by considering
special toric compactifications which make their singular structure apparent.
This gives a precise description of their convergence domain, refining results
of Nilsson, Passare, Berkesch and Forsg\aa rd. We also reformulate the
geometric sector decomposition approach of Kaneko and Ueda in terms of these
compactifications. Specializing to the case of Feynman integrals in the
parametric representation, we construct multiple such compactifications given
by certain systems of subgraphs. As particular cases, we recover the sector
decompositions of Hepp, Speer and Smirnov, as well as the iterated blow-up
constructions of Brown and Bloch-Esnault-Kreimer. A fundamental role is played
by the Newton polytope of the product of the Symanzik polynomials, which we
show to be a generalized permutahedron for generic kinematics. As an
application, we review two approaches to dimensional regularization of Feynman
integrals, based on sector decomposition and on analytic continuation.
