The extremals of Stanley's inequalities for partially ordered sets

Summary: This article provides elementary self-contained proofs of the strong Mason conjecture and the classical Alexandrov-Fenchel inequality, recently proven by Anari et al. (2018) and Branden and Huh (2020) respectively. Both proofs utilize the combinatorial atlas technology introduced by Chan and Pak (2021). Additionally, a formal relationship between combinatorial atlases and Lorentzian polynomials is presented.