The chain polytope of a finite partially ordered set 
is the convex polytope
 |
(1)
|
It suffices to impose the inequalities for maximal chains. The vertices are the coordinate vectors of the indicator functions of antichains, including the empty antichain (Stanley 1986).
If 
is an 
-element chain, then 
is the simplex defined by 
and 
. If 
is an 
-element antichain, it is the
unit hypercube.
Stanley's (1986) transfer map from the order polytope to the chain polytope is
 |
(2)
|
This is a piecewise linear bijection. The two polytopes have the same Ehrhart polynomial, volume, and number of vertices.
Their 2-dimensional faces are triangles or quadrilaterals. The quadrilateral counts agree, while the triangular count for the chain polytope is at least that for the order polytope (Freij-Hollanti et al. 2026).
For a partially ordered set 
of 
elements containing the five-element obstruction with two
incomparable elements below one element and two incomparable elements above it, Goel
(2026) proves
 |
(3)
|
for 
. Here the obstruction is an induced partially ordered set, and 
counts 
-dimensional faces. For an arbitrary
-element partially
ordered set, equality of the face counts in this range characterizes unimodular
equivalence with the order polytope.
