Two-Dimensional Faces of Order and Chain Polytopes

We give an explicit combinatorial description of the two-dimensional faces of both the order polytope O(P) and the chain polytope C(P) of a partially ordered set P. Using these descriptions, we show that for any P, C(P) has equally many square faces, and at least as many triangular faces, as O(P) does. Moreover, the inequality is shown to be strict except when O(P) and C(P) are unimodularly equivalent. This proves the case $i=2$ of a conjecture by Hibi and Li.