Dually Lorentzian Polynomials

We introduce and study a notion of dually Lorentzian polynomials, and show that if s is non-zero and dually Lorentzian then the operator s(∂ x₁,… ,∂ xₙ): R[x₁,… ,xₙ] → R[x₁,… ,xₙ] s ( ∂ x 1 , … , ∂ x n ) : R [ x 1 , … , x n ] → R [ x 1 , … , x n ] preserves (strictly) Lorentzian polynomials. From this we conclude that any theory that admits a mixed Alexandrov-Fenchel inequality also admits a generalized Alexandrov-Fenchel inequality involving dually Lorentzian polynomials. As such we deduce generalized Alexandrov-Fenchel inequalities for mixed discriminants, for integrals of Kähler classes, for mixed volumes, and in the theory of valuations.