We give elementary self-contained proofs of the strong Mason conjecture recently proved by Anari at. al. (arXiv:1811.01600) and Brändén--Huh (arXiv:1902.03719), and of the classical Alexandrov--Fenchel inequality. Both proofs use the combinatorial atlas technology recently introduced by the authors (arXiv:2110.10740). We also give a formal relationship between combinatorial atlases and Lorentzian polynomials.