Toric polar maps, normal crossings and mixed volumes

Given a hypersurface in a complex projective space, we prove that the Chern–Schwartz–MacPherson class of a certain open set, namely the complement of the union of the hypersurface and the coordinate hyperplanes, is given (up to sign) by the multidegrees of associated toric polar map, in two particular cases: normal crossings with the coordinate hyperplanes and a nondegeneracy condition with respect to the Newton polytope. In the latter case, we also recover the multidegrees from mixed volumes.