Dually Lorentzian Polynomials

Lorentzian polynomials, recently introduced by Brändén and Huh, have coefficients that satisfy a form of log-concavity, and have been used to prove, reprove, and conjecture various combinatorial statements coming from convex geometry, representation theory, and the theory of matroids. Accepting that being Lorentzian is a useful concept, it is natural to ask for differential operators (with constant coefficients) that preserve this property. This leads to the notion of dually Lorentzian polynomials. As an application of this observation, I will show how dually Lorentzian polynomials give rise to generalisations of the Alexandrov-Fenchel inequality in convex geometry.

This is joint work with Julius Ross and Thomas Wannerer.