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 $$s({\mathrm{\partial }}_{x_1},\dots ,{\mathrm{\partial }}_{x_n}):\mathbb{R}[x_1,\dots ,x_n]\to \mathbb{R}[x_1,\dots ,x_n]$$ 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.