Cutting convex polytopes by hyperplanes

Cutting a polytope is a very natural way to produce new classes of interesting polytopes. Moreover, it has been very enlightening to explore which algebraic and combinatorial properties of the orignial polytope are hereditary to its subpolytopes obtained by a cut. In this work, we put our attention to all the seperating hyperplanes for some given polytope (integral and convex) and study the existence and classification of such hyperplanes. We prove the exitence of seperating hyperplanes for the order and chain polytopes for any finite posets that are not a single chain; prove there are no such hyperplanes for any Birkhoff polytopes. Moreover, we give a complete seperating hyperplane classification for the unit cube and its subpolytopes obtained by one cut, together with some partial classification results for order and chain polytopes.