  Stanley's inequalities for partially ordered sets establish important
log-concavity relations for sequences of linear extensions counts. Their
extremals however, i.e., the equality cases of these inequalities, were until
now poorly understood with even conjectures lacking. In this work, we solve
this problem by providing a complete characterization of the extremals of
Stanley's inequalities. Our proof is based on building a new ``dictionary"
between the combinatorics of partially ordered sets and the geometry of convex
polytopes, which captures their extremal structures.
