Abstract
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 original polytope are hereditary to its subpolytopes obtained by a cut. In this work, we devote our attention to all the separating hyperplanes for some given polytope (integral and convex) and study the existence and classification of such hyperplanes. We prove the existence of separating hyperplanes for the order and chain polytopes for any finite posets that are not a single chain, and prove there are no such hyperplanes for any Birkhoff polytopes. Moreover, we give a complete separating hyperplane classification for the unit cube and its subpolytopes obtained by one cut, together with some partial classification results for order and chain polytopes.
Highlights
Let P ⊂ Rn be a convex polytope of dimension d and ∂P its boundary
When H ∩ (P \ ∂P ) 6= ∅, it follows that H cuts P if and only if, for each edge e = conv({v, v0 }) of P, where v and v0 are vertices of P, and not e ⊂ H, one has H ∩ e ⊂ {v, v0 }
We focus our study on the following classes of polytopes: The unit cube and its subpolytopes cut by one hyperplane, order and chain polytopes, and Birkhoff polytopes
Summary
Let P ⊂ Rn be a convex polytope of dimension d and ∂P its boundary. If H ⊂ Rn is a hyperplane, we write H(+) and H (−) for the closed half-spaces of Rn with H (+) ∩ H(−) = H. A similar class of interesting polytopes obtained from cutting permutahedrons and in general any graphical zonotopes are studied in [4]. It follows that a hyperplane H ⊂ Rn is a separating hyperplane of P if and only if each of the subpolytopes P ∩ H (+) and P ∩ H (−) is integral of dimension d. The study of existence and classification for any general internal convex polytopes can be very hard. We focus our study on the following classes of polytopes: The unit cube and its subpolytopes cut by one hyperplane, order and chain polytopes, and Birkhoff polytopes. We give a complete separating hyperplane classification for the unit cube and its subpolytopes cut by one hyperplane (Section 1), together with partial classification results for order and chain polytopes (Section 2)
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