Linear extensions of finite posets are fundamental combinatorial objects, which generalize increasing trees and standard Young tableaux. Stanley's inequality states that the number of linear extensions of a poset with value k at a fixed posed element, is log-concave in k. This was famously proved by Stanley in 1981 as an application of the Alexandrov-Fenchel inequalities in convex geometry. In the past few years this inequality received much attention, both in connection to convex geometry by Shenfeld and van Handel, and as an application of the combinatorial atlas technology by S.H. Chan and myself. In the first part of the talk, I will define and discuss linear extensions in the context of combinatorial inequalities. I will then review recent work on equality conditions of Stanley's inequality and applications showing nonexistence of combinatorial interpretations for their defect. The talk will assume no prior knowledge of the subject.