The order polytope of a finite partially ordered set 
is the convex polytope
![O(P)={x in [0,1]^P:x_p<=x_q whenever p<=_Pq}.](/images/equations/OrderPolytope/NumberedEquation1.svg) |
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 ![[0,1]^n](/images/equations/OrderPolytope/Inline7.svg)
.
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.
