On facet gaps of order and chain polytopes

We discuss three questions from a recent paper of Bhandari, Cunningham, Morrell, Oh and Smith. We obtain an exact local formula for the facet gap between the order and the chain polytope of a finite poset. This gives a classification of the case $\gap(P)=2$ in terms of star elements. For marked chain--order polytopes, the same local weights determine the facet differences between all admissible decompositions.