Order Polytope

The order polytope of a finite partially ordered set is the convex polytope

Its dimension is . Its vertices are the coordinate vectors of the indicator functions of upward-closed subsets of , including the empty subset (Stanley 1986). The complements of these subsets are partial order ideals.

For a chain , the inequalities are , giving a simplex. For an -element antichain, the order polytope is the unit hypercube .

The order polytope and chain polytope have the same Ehrhart polynomial and volume , where and counts the linear extensions of (Stanley 1986).

Freij-Hollanti et al. (2026) prove that the chain polytope has the same number of quadrilateral 2-dimensional faces and at least as many triangular 2-dimensional faces. Equality in the triangular counts holds iff the two polytopes are related by an affine map with integer and . The paper calls quadrilateral faces "square faces" without requiring Euclidean squares.

The order polytope of the alternating partially ordered set on elements with relations is integrally equivalent to the unit-flow polytope of the zigzag directed graph on vertices (Brunner and Hanusa 2026).

Goel (2026, Theorem 4.1) proves a sharp quantitative form of the comparison with the chain polytope. Let have elements and contain, as an induced partially ordered set, an element above two incomparable elements and below two other incomparable elements. For ,

where counts the -dimensional faces. For each , examples attain all these bounds simultaneously. For an arbitrary -element partially ordered set, equality of the two face counts for any such holds iff the two polytopes are unimodularly equivalent. The analogous implication for is false, since the numbers of edges agree even for inequivalent pairs. Goel (2026) reports an AI-developed Lean proof that assumes two results from Fourier's face-comparison argument.