Lorentzian polynomials are a fascinating class of real polynomials with many applications. Their definition is specific to the nonnegative orthant. Following recent work, we examine Lorentzian polynomials on proper convex cones. For a self-dual cone \({\mathcal {K}}\) we find a connection between \({\mathcal {K}}\) -Lorentzian polynomials and \({\mathcal {K}}\) -positive linear maps, which were studied in the context of the generalized Perron-Frobenius theorem. We find that as the cone \({\mathcal {K}}\) varies, even the set of quadratic \({\mathcal {K}}\) -Lorentzian polynomials can be difficult to understand algorithmically. We also show that, just as in the case of the nonnegative orthant, \({\mathcal {K}}\) -Lorentzian and \({\mathcal {K}}\) -completely log-concave polynomials coincide.
