Faces of 2-Dimensional Simplex of Order and Chain Polytopes

Abstract

Each of the descriptions of vertices, edges, and facets of the order and chain polytope of a finite partially ordered set are well known. In this paper, we give an explicit description of faces of 2-dimensional simplex in terms of vertices. Namely, it will be proved that an arbitrary triangle in 1-skeleton of the order or chain polytope forms the face of 2-dimensional simplex of each polytope. These results mean a generalization in the case of 2-faces of the characterization known in the case of edges.

1. Introduction

The combinatorial structure of the order polytope $\mathcal{O}\left(P\right)$ and the chain polytope $\mathcal{C}\left(P\right)$ of a finite poset (partially ordered set) P is explicitly discussed in [1]. Moreover, in [2], the problem when the order polytope $\mathcal{O}\left(P\right)$ and the chain polytope $\mathcal{C}\left(P\right)$ are unimodularly equivalent is solved. It is also proved that the number of edges of the order polytope $\mathcal{O}\left(P\right)$ is equal to that of the chain polytope $\mathcal{C}\left(P\right)$ in [3]. In the present paper we give an explicit description of faces of 2-dimensional simplex of $\mathcal{O}\left(P\right)$ and $\mathcal{C}\left(P\right)$ in terms of vertices. In other words, we show that triangles in 1-skeleton of $\mathcal{O}\left(P\right)$ or $\mathcal{C}\left(P\right)$ are in one-to-one correspondence with faces of 2-dimensional simplex of each polytope. These results are a direct generalizations of [4] (Lemma 4, Lemma 5).

2. Definition and Known Results

Let $P=\{x_1,\cdots ,x_d\}$ be a finite poset. To each subset $W\subset P$, we associate $\rho \left(W\right)={\sum }_{i\in W}{\mathbf{e}}_i\in {\mathbb{R}}^d$, where ${\mathbf{e}}_1,\cdots ,{\mathbf{e}}_d$ are the canonical unit coordinate vectors of ${\mathbb{R}}^d$. In particular $\rho (\varnothing )$ is the origin of ${\mathbb{R}}^d$. A poset ideal of P is a subset I of P such that, for all $x_i$ and $x_j$ with $x_i\in I$ and $x_j\le x_i$, one has $x_j\in I$. An antichain of P is a subset A of P such that $x_i$ and $x_j$ belonging to A with $i\ne j$ are incomparable. The empty set ∅ is a poset ideal as well as an antichain of P. We say that $x_j$covers$x_i$ if $x_i<x_j$ and $x_i<x_k<x_j$ for no $x_k\in P$. A chain $x_{j_1}<x_{j_2}<\cdots <x_{j_{\ell }}$ of P is called saturated if $x_{j_q}$ covers $x_{j_{q-1}}$ for $1<q\le \ell$. A maximal chain is a saturated chain such that $x_{j_1}$ is a minimal element and $x_{j_{\ell }}$ is a maximal element of the poset. The rank of P is $♯\left(C\right)-1$, where C is a chain with maximum length of P.

The order polytope of P is the convex polytope $\mathcal{O}\left(P\right)\subset {\mathbb{R}}^d$ which consists of those $(a_1,\cdots a_d)\in {\mathbb{R}}^d$ such that $0\le a_i\le 1$ for every $1\le i\le d$ together with

$$a_i\ge a_j$$

if $x_i\le x_j$ in P.

The chain polytope of P is the convex polytope $\mathcal{C}\left(P\right)\subset {\mathbb{R}}^d$ which consists of those $(a_1,\cdots ,a_d)\in {\mathbb{R}}^d$ such that $a_i\ge 0$ for every $1\le i\le d$ together with

$$a_{i_i}+a_{i_2}+\cdots +a_{i_k}\le 1$$

for every maximal chain $x_{i_1}<x_{i_2}<\cdots <x_{i_k}$ of P.

One has $\dim \mathcal{O}\left(P\right)=\dim \mathcal{C}\left(P\right)=d$. The vertices of $\mathcal{O}\left(P\right)$ is those $\rho \left(I\right)$ for which I is a poset ideal of P ([1] (Corollary1.3)) and the vertices of $\mathcal{C}\left(P\right)$ is those $\rho \left(A\right)$ for which A is an antichain of P ([1] (Theorem2.2)). It then follows that the number of vertices of $\mathcal{O}\left(P\right)$ is equal to that of $\mathcal{C}\left(P\right)$. Moreover, the volume of $\mathcal{O}\left(P\right)$ and that of $\mathcal{C}\left(P\right)$ are equal to $e\left(P\right)/d!$, where $e\left(P\right)$ is the number of linear extensions of P ([1] (Corollary4.2)). It also follows from [1] that the facets of $\mathcal{O}\left(P\right)$ are the following:

• $x_i=0$, where $x_i\in P$ is maximal;
• $x_j=1$, where $x_j\in P$ is minimal;
• $x_i=x_j$, where $x_j$ covers $x_i$,

and that the facets of $\mathcal{C}\left(P\right)$ are the following:

• $x_i=0$, for all $x_i\in P$;
• $x_{i_1}+\cdots +x_{i_k}=1$, where $x_{i_1}<\cdots <x_{i_k}$ is a maximal chain of P.

In [4] a characterization of edges of $\mathcal{O}\left(P\right)$ and those of $\mathcal{C}\left(P\right)$ is obtained. Recall that a subposet Q of finite poset P is said to be connected in P if, for each x and y belonging to Q, there exists a sequence $x=x_0,x_1,\cdots ,x_s=y$ with each $x_i\in Q$ for which $x_{i-1}$ and $x_i$ are comparable in P for each $1\le i\le s$.

Lemma 1

([4] (Lemma 4, Lemma 5)) Let P be a finite poset.

1.

Let I and J be poset ideals of P with $I\ne J$. Then the convex hull of $\left\{\rho \right(I),\rho (J\left)\right\}$ forms an edge of $\mathcal{O}\left(P\right)$ if and only if $I\subset J$ and $J\setminus I$ is connected in P.

2.

Let A and B be antichains of P with $A\ne B$. Then the convex hull of $\left\{\rho \right(A),\rho (B\left)\right\}$ forms an edge of $\mathcal{C}\left(P\right)$ if and only if $(A\setminus B)\cup (B\setminus A)$ is connected in P.

3. Faces of 2-Dimensional Simplex

Using Lemma 1, we show the following description of faces of 2-dimensional simplex.

Theorem 1.

Let P be a finite poset. Let I, J, and K be pairwise distinct poset ideals of P. Then the convex hull of $\left\{\rho \right(I),\rho (J),\rho (K\left)\right\}$ forms a 2-face of $\mathcal{O}\left(P\right)$ if and only if $I\subset J\subset K$ and $K\setminus I$ is connected in P.

Proof. 

(“Only if”) If the convex hull of $\left\{\rho \right(I),\rho (J),\rho (K\left)\right\}$ forms a 2-face of $\mathcal{O}\left(P\right)$, then the convex hulls of $\left\{\rho \right(I),\rho (J\left)\right\}$, $\left\{\rho \right(J),\rho (K\left)\right\}$, and $\left\{\rho \right(I),\rho (K\left)\right\}$ form edges of $\mathcal{O}\left(P\right)$. It then follows from Lemma 1 that $I\subset J\subset K$ and $K\setminus I$ is connected in P.

(“If”) Suppose that the convex hull of $\left\{\rho \right(I),\rho (J),\rho (K\left)\right\}$ has dimension 1. Then there exists a line passing through the lattice points $\rho \left(I\right),\rho \left(J\right)$, and $\rho \left(K\right)$. Hence $\rho \left(I\right),\rho \left(J\right)$, and $\rho \left(K\right)$ cannot be vertices of $\mathcal{O}\left(P\right)$. Thus the convex hull of $\left\{\rho \right(I),\rho (J),\rho (K\left)\right\}$ has dimension 2.

Let $P=\{x_1,\cdots ,x_d\}$. If there exists a maximal element $x_i$ of P not belonging to $I\cup J\cup K$, then the convex hull of $\left\{\rho \right(I),\rho (J),\rho (K\left)\right\}$ lies in the facet $x_i=0$. If there exists a minimal element $x_j$ of P belonging to $I\cap J\cap K$, then the convex hull of $\left\{\rho \right(I),\rho (J),\rho (K\left)\right\}$ lies in the facet $x_j=1$. Hence, working with induction on $d(\ge 2)$, we may assume that $I\cup J\cup K=P$ and $I\cap J\cap K=\varnothing$. Suppose that $\varnothing =I\subset J\subset K=P$ and $K\setminus I=P$ is connected.

Case 1.$♯\left(J\right)=1$.

Let $J=\left\{x_i\right\}$ and $P^{\prime }=P\setminus \left\{x_i\right\}$. Then $P^{\prime }$ is a connected poset. Let $x_{i_1},\cdots ,x_{i_q}$ be the maximal elements of P and ${\mathcal{A}}_{ij}=\{y\in P^{\prime }|y<x_{i_j}\}$ , where $1\le j\le q$. Then we write

$$b_k=\left\{\begin{array}{cc}♯\left(\left\{i_j\right|x_k\in {\mathcal{A}}_{ij}\}\right)\hfill & \mathrm{if}k\notin \{i_1,\cdots ,i_q,i\}\hfill \\ 0\hfill & \mathrm{if}k=i\hfill \\ -♯\left({\mathcal{A}}_{ij}\right)\hfill & \mathrm{if}k\in \{i_1,\cdots ,i_q\}\hfill \end{array}\right..$$

We then claim that the hyperplane $\mathcal{H}$ of ${\mathbb{R}}^d$ defined by the equation $h\left(\mathbf{x}\right)={\sum }_{k=1}^d{b}_k{x}_k=0$ is a supporting hyperplane of $\mathcal{O}\left(P\right)$ and that $\mathcal{H}\cap \mathcal{O}\left(P\right)$ coincides with the convex hull of $\left\{\rho \right(\varnothing ),\rho (J),\rho (P\left)\right\}$. Clearly $h\left(\rho \right(\varnothing \left)\right)=h\left(\rho \right(P\left)\right)=0$ and $h\left(\rho \left(J\right)\right)=b_i=0$. Let I be a poset ideal of P with $I\ne \varnothing$, $I\ne P$ and $I\ne J$. We have to prove that $h\left(\rho \right(I\left)\right)>0$. To simplify the notation, suppose that $I\cap \{x_{i_1},\cdots ,x_{i_q}\}=\{x_{i_1},\cdots ,x_{i_r}\}$, where $0\le r<q$. If $r=0$, then $h\left(\rho \right(J\left)\right)>0$. Let $1\le r<q$, $I^{\prime }=I\setminus \left\{x_i\right\}$, and $K={\bigcup }_{j=1}^r\left({\mathcal{A}}_{i_j}\cup \left\{x_{i_j}\right\}\right)$. Then $I^{\prime }$ and K are poset ideals of P and $h\left(\rho \left(K\right)\right)\le h\left(\rho \left(I^{\prime }\right)\right)=h\left(\rho \left(I\right)\right)$. We claim $h\left(\rho \right(K\left)\right)>0$. One has $h\left(\rho \right(K\left)\right)\ge 0$. Moreover, $h\left(\rho \right(K\left)\right)=0$ if and only if no $z\in K$ belongs to ${\mathcal{A}}_{i_{r+1}}\cup \cdots \cup {\mathcal{A}}_{i_q}$. Now, since $P^{\prime }$ is connected, it follows that there exists $z\in K$ with $z\in {\mathcal{A}}_{i_{r+1}}\cup \cdots \cup {\mathcal{A}}_{i_q}$. Hence $h\left(\rho \right(K\left)\right)>0$. Thus $h\left(\rho \right(I\left)\right)>0$.

Case 2.$♯\left(J\right)=d-1$.

Let $P\setminus J=\left\{x_i\right\}$ and $P^{\prime }=P\setminus \left\{x_i\right\}$. Then $P^{\prime }$ is a connected poset. Thus we can show the existence of a supporting hyperplane of $\mathcal{O}\left(P\right)$ which contains the convex hull of $\left\{\rho \right(\varnothing ),\rho (J),\rho (P\left)\right\}$ by the same argument in Case 1.

Case 3.$2\le ♯\left(J\right)\le d-2$.

To simplify the notation, suppose that $J=\{x_1,\cdots ,x_{\ell }\}$. Then $P\setminus J=\{x_{\ell +1},\cdots ,x_d\}$. Since J and $P\setminus J$ are subposets of P, these posets are connected. Let $x_{i_1},\cdots ,x_{i_q}$ be the maximal elements of J and $x_{i_{q+1}},\cdots ,x_{i_{q+r}}$ the maximal elements of $P\setminus J$. Then we write

$${\mathcal{A}}_{ij}=\left\{\begin{array}{cc}\{y\in J|y<x_{i_j}\}\hfill & \mathrm{if}1\le j\le q\hfill \\ \{y\in P\setminus J|y<x_{i_j}\}\hfill & \mathrm{if}q+1\le j\le r\hfill \end{array}\right.$$

and

$$b_k=\left\{\begin{array}{cc}♯\left(\left\{i_j\right|x_i\in {\mathcal{A}}_{ij}\}\right)\hfill & \mathrm{if}k\notin \{i_1,\cdots ,i_q,i_{q+1},\cdots ,i_{q+r}\}\hfill \\ -♯\left({\mathcal{A}}_{ij}\right)\hfill & \mathrm{if}k\in \{i_1,\cdots ,i_q,i_{q+1},\cdots ,i_{q+r}\}\hfill \end{array}\right..$$

We then claim that the hyperplane $\mathcal{H}$ of ${\mathbb{R}}^d$ defined by the equation $h\left(\mathbf{x}\right)={\sum }_{k=1}^d{b}_k{x}_k=0$ is a supporting hyperplane of $\mathcal{O}\left(P\right)$ and $\mathcal{H}\cap \mathcal{O}\left(P\right)$ coincides with the convex hull of $\left\{\rho \right(\varnothing ),\rho (J),\rho (P\left)\right\}$. Clearly $h\left(\rho \right(\varnothing \left)\right)=h\left(\rho \right(J\left)\right)=h\left(\rho \right(P\setminus J\left)\right)=0$, then $h\left(\rho \right(P\left)\right)=h\left(\rho \right(J\left)\right)+h\left(\rho \right(P\setminus J\left)\right)=0$. Let I be a poset ideal of P with $I\ne \varnothing$, $I\ne P$ and $I\ne J$. What we must prove is $h\left(\rho \right(I\left)\right)>0$.

If $I\subset J$, then I is a poset ideal of J. To simplify the notation, suppose that $I\cap \{x_{i_1},\cdots ,x_{i_q}\}=\{x_{i_1},\cdots ,x_{i_s}\}$ , where $0\le s<q$. If $s=0$, then $h\left(\rho \right(I\left)\right)>0$. Let $1\le s<q$, $K={\bigcup }_{j=1}^s\left({\mathcal{A}}_{i_j}\cup \left\{x_{i_j}\right\}\right)$. Then K is a poset ideal of J and $h\left(\rho \right(K\left)\right)\le h\left(\rho \right(I\left)\right)$. Thus we can show $h\left(\rho \right(K\left)\right)>0$ by the same argument in Case 1 (Replace r with s and $P^{\prime }$ with J).

If $J\subset I$, then $I\setminus J$ is a poset ideal of $P\setminus J$. To simplify the notation, suppose that $(I\setminus J)\cap \{x_{i_{q+1}},\cdots ,x_{i_{q+r}}\}=\{x_{i_{q+1}},\cdots ,x_{i_{q+t}}\}$ , where $0\le t<r$. If $t=0$, then $h\left(\rho \right(I\left)\right)=h\left(\rho \right(J\left)\right)+h\left(\rho \right(I\setminus J\left)\right)=h\left(\rho \right(I\setminus J\left)\right)>0$. Let $1\le t<r$, $K={\bigcup }_{j=q+1}^{q+t}\left({\mathcal{A}}_{i_j}\cup \left\{x_{i_j}\right\}\right)$. Then K is a poset ideal of $P\setminus J$ and $h\left(\rho \right(K\left)\right)\le h\left(\rho \right(I\setminus J\left)\right)=h\left(\rho \right(I\left)\right)$. Thus we can show $h\left(\rho \right(K\left)\right)>0$ by the same argument in Case 1 (Replace r with $q+t$, q with $q+r$ and $P^{\prime }$ with $P\setminus J$). Consequently, $h\left(\rho \right(I\left)\right)>0$, as desired. □

Let $A▵B$ denote the symmetric difference of the sets A and B, that is $A▵B=(A\setminus B)\cup (B\setminus A)$.

Theorem 2.

Let P be a finite poset. Let A, B, and C be pairwise distinct antichains of P. Then the convex hull of $\left\{\rho \right(A),\rho (B),\rho (C\left)\right\}$ forms a 2-face of $\mathcal{C}\left(P\right)$ if and only if $A▵B$, $B▵C$ and $C▵A$ are connected in P.

Proof. 

(“Only if”) If the convex hull of $\left\{\rho \right(A),\rho (B),\rho (C\left)\right\}$ forms a 2-face of $\mathcal{C}\left(P\right)$, then the convex hulls of $\left\{\rho \right(A),\rho (B\left)\right\}$, $\left\{\rho \right(B),\rho (C\left)\right\}$, and $\left\{\rho \right(A),\rho (C\left)\right\}$ form edges of $\mathcal{C}\left(P\right)$. It then follows from Lemma 1 that $A▵B$, $B▵C$ and $C▵A$ are connected in P.

(“If”) Suppose that the convex hull of $\left\{\rho \right(A),\rho (B),\rho (C\left)\right\}$ has dimension 1. Then there exists a line passing through the lattice points $\rho \left(A\right)$, $\rho \left(B\right)$, and $\rho \left(C\right)$. Hence $\rho \left(A\right)$, $\rho \left(B\right)$, and $\rho \left(C\right)$ cannot be vertices of $\mathcal{C}\left(P\right)$. Thus the convex hull of $\left\{\rho \right(A),\rho (B),\rho (C\left)\right\}$ has dimension 2.

Let $P=\{x_1,\cdots ,x_d\}$. If $A\cup B\cup C\ne P$ and $x_i\notin A\cup B\cup C$, then the convex hull of $\left\{\rho \right(A),\rho (B),\rho (C\left)\right\}$ lies in the facet $x_i=0$. Furthermore, if $A\cup B\cup C=P$ and $A\cap B\cap C\ne \varnothing$, then $x_j\in A\cap B\cap C$ is isolated in P and $x_j$ itself is a maximal chain of P. Thus the convex hull of $\left\{\rho \right(A),\rho (B),\rho (C\left)\right\}$ lies in the facet $x_j=1$. Hence, working with induction on $d(\ge 2)$, we may assume that $A\cup B\cup C=P$ and $A\cap B\cap C=\varnothing$. As stated in the proof of [3] ([Theorem 2.1]), if $A▵B$ is connected in P, then A and B satisfy either (i) $B\subset A$ or (ii) $y<x$ whenever $x\in A$ and $y\in B$ are comparable. Hence, we consider the following three cases:

(a) If $B\subset A$, then $A▵B=A\setminus B$ is connected in P, and thus $♯(A\setminus B)=1$. Let $A\setminus B=\left\{x_k\right\}$. If $C\cap A\ne \varnothing$, then $C\cap A=\left\{x_k\right\}$, since $A\cap B\cap C=C\cap B=\varnothing$. Namely $x_k$ is isolated in P. Hence $B▵C=B\cup C=A\cup B\cup C=P$ cannot be connected. Thus $C\cap A=\varnothing$. In this case, we may assume $z<x$ if $x\in A$ and $z\in C$ are comparable. Furthermore, P has rank 1.

(b) If $B\not\subset A$ and $B\cap A\ne \varnothing$, then we may assume $y<x$ if $x\in A$ and $y\in B$ are comparable. If $C\subset B$ with $C\cap A\cap B=\varnothing$, then as stated in (a), $C▵A$ cannot be connected. Since $C\not\subset B$, we may assume $z<y$ if $y\in B$ and $z\in C$ are comparable. If $C\cap B\ne \varnothing$, then $C\cap A=\varnothing$ and P has rank 1 or 2. Similarly, if $C\cap B=\varnothing$, then $C\cap A=\varnothing$ and P has rank 2.

(c) Let $B\not\subset A$ and $B\cap A=\varnothing$. We may assume that if $x\in A$ and $y\in B$ are comparable, then $y<x$. If $C\subset B$, then we regard this case as equivalent to (a). Let $C\not\subset B$. We may assume $z<y$ if $y\in B$ and $z\in C$ are comparable. Moreover, if $C\cap B\ne \varnothing$, then we regard this case as equivalent to (b). If $C\cap B=\varnothing$, then $C\cap A=\varnothing$ and P has rank 2.

Consequently, there are five cases as regards antichains for $\mathcal{C}\left(P\right)$.

Case 1.$B\subset A$, $C\cap A=\varnothing$, and $C\cap B=\varnothing$.

For each $x_i\in B$ we write $b_i$ for the number of elements $z\in C$ with $z<x_i$. For each $x_j\in C$ we write $c_j$ for the number of elements $y\in B$ with $x_j<y$. Let $a_k=0$ for $A\setminus B=\left\{x_k\right\}$. Clearly ${\sum }_{x_i\in B}{b}_i={\sum }_{x_j\in C}{c}_j=q$, where q is the number of pairs $(y,z)$ with $y\in B$, $z\in C$ and $z<y$. Let $h\left(\mathbf{x}\right)={\sum }_{x_i\in B}{b}_i{x}_i+{\sum }_{x_j\in C}{c}_j{x}_j+a_k{x}_k$ and let $\mathcal{H}$ be the hyperplane of ${\mathbb{R}}^d$ defined by $h\left(\mathbf{x}\right)=q$. Then $h\left(\rho \right(A\left)\right)=h\left(\rho \right(B\left)\right)=h\left(\rho \right(C\left)\right)=q$. We claim that, for any antichain D of P with $D\ne A$, $D\ne B$, and $D\ne C$, one has $h\left(\rho \right(D\left)\right)<q$. Let $D=B_1\cup C_1$ or $D=\left\{x_k\right\}\cup C_1$ with $B_1⊊B$ and $C_1⊊C$. Suppose $D=B_1\cup C_1$. Since $B▵C$ is connected and since D is an antichain of P, it follows that ${\sum }_{x_i\in B_1}{b}_i+{\sum }_{x_j\in C_1}{c}_j<q$. Thus $h\left(\rho \right(D\left)\right)<q.$ Suppose that $D=\left\{x_k\right\}\cup C_1$. It follows that ${\sum }_{x_j\in C_1}{c}_j+a_k={\sum }_{x_j\in C_1}{c}_j<{\sum }_{x_j\in C}{c}_j=q.$ Thus $h\left(\rho \right(D\left)\right)<q.$

Case 2.$B\not\subset A$, $B\cap A\ne \varnothing$, $C\not\subset B$, $C\cap B\ne \varnothing$, $C\cap A=\varnothing$, and P has rank 1.

We define four numbers as follows:

$$\begin{array}{cc}\hfill {\alpha }_i& =♯\left(\{y\in B\setminus A|y<x_i,x_i\in A\setminus B\}\right);\hfill \\ \hfill {\gamma }_j& =♯\left(\{x\in A\setminus B|x_j<x,x_j\in B\setminus A\}\right);\hfill \\ \hfill {\alpha }_k& =♯\left(\{z\in C\setminus B|z<x_k,x_k\in B\setminus C\}\right);\hfill \\ \hfill {\gamma }_{\ell }& =♯\left(\{y\in B\setminus C|x_{\ell }<y,x_{\ell }\in C\setminus B\}\right).\hfill \end{array}$$

Since P has rank 1, $B\subset A\cup C=P$. It follows that $A=(A\setminus B)\cup (B\setminus C)$, $C=(B\setminus A)\cup (C\setminus B)$. Then

$$\begin{array}{cc}\hfill \sum _{x_s\in A}{\alpha }_s=\sum _{x_i\in A\setminus B}{\alpha }_i+\sum _{x_k\in B\setminus C}{\alpha }_k& =q;\hfill \\ \hfill \sum _{x_j\in B\setminus A}{\gamma }_j+\sum _{x_k\in B\setminus C}{\alpha }_k& =q;\hfill \\ \hfill \sum _{x_u\in C}{\gamma }_u=\sum _{x_j\in B\setminus A}{\gamma }_j+\sum _{x_{\ell }\in C\setminus B}{\gamma }_{\ell }& =q,\hfill \end{array}$$

where $q_1$ is the number of pairs $(x,y)$ with $x\in A\setminus B$, $y\in B\setminus A$ and $y<x$, $q_2$ is the number of pairs $(y,z)$ with $y\in B\setminus C$, $z\in C\setminus B$ and $z<y$, and $q=q_1+q_2$. Let

$$\begin{array}{cc}\hfill h\left(\mathbf{x}\right)& =\sum _{x_s\in A}{\alpha }_s{x}_s+\sum _{x_u\in C}{\gamma }_u{x}_u\hfill \\ \hfill & =\sum _{x_i\in A\setminus B}{\alpha }_i{x}_i+\left(\sum _{x_j\in B\setminus A}{\gamma }_j{x}_j+\sum _{x_k\in B\setminus C}{\alpha }_k{x}_k\right)+\sum _{x_{\ell }\in C\setminus B}{\gamma }_{\ell }{x}_{\ell }\hfill \end{array}$$

and $\mathcal{H}$ the hyperplane of ${\mathbb{R}}^d$ defined by $h\left(\mathbf{x}\right)=q$. Then $h\left(\rho \right(A\left)\right)=h\left(\rho \right(B\left)\right)=h\left(\rho \right(C\left)\right)=q$. We claim that, for any antichain D of P with $D\ne A$, $D\ne B$ and $D\ne C$, one has $h\left(\rho \right(D\left)\right)<q$. Let $D=D_1\cup D_2$ with $D_1$ is an antichain of $A▵B$ and $D_2$ is an antichain of $B▵C$. Since $A▵B$, $B▵C$ are connected, it follows that $h\left(\rho \left(D_1\right)\right)<q_1$ and $h\left(\rho \left(D_2\right)\right)<q_2$. Thus $h\left(\rho \left(D\right)\right)=h\left(\rho \left(D_1\right)\right)+h\left(\rho \left(D_2\right)\right)<q_1+q_2=q.$

Case 3.$B\not\subset A$, $B\cap A\ne \varnothing$, $C\not\subset B$, $C\cap B\ne \varnothing$, $C\cap A=\varnothing$, and P has rank 2.

For each $x_i\in P$ we write $c\left(i\right)$ for the number of maximal chains, which contain $x_i$. Let q be the number of maximal chains in P. Since each $x_i\in A$ is maximal element and each $x_k\in C$ is minimal element, ${\sum }_{x_i\in A}c\left(i\right)={\sum }_{x_k\in C}c\left(k\right)=q$. Then

$$\begin{array}{cc}\hfill \sum _{x_j\in B}c\left(j\right)& =\sum _{x_s\in B\cap A}c\left(s\right)+\sum _{x_t\in B\cap C}c\left(t\right)+\sum _{x_u\in B\setminus (A\cup C)}c\left(u\right)\hfill \\ \hfill & =\sum _{x_s\in B\cap A}c\left(s\right)+\sum _{x_t\in B\cap C}c\left(t\right)+\left(\sum _{x_v\in A\setminus B}c\left(v\right)-\sum _{x_t\in B\cap C}c\left(t\right)\right)\hfill \\ \hfill & =\sum _{x_i\in A}c\left(i\right)=q.\hfill \end{array}$$

Let $h\left(\mathbf{x}\right)={\sum }_{x_i\in P}c\left(i\right)x_i$ and $\mathcal{H}$ the hyperplane of ${\mathbb{R}}^d$ defined by $h\left(\mathbf{x}\right)=q$. Then $h\left(\rho \right(A\left)\right)=h\left(\rho \right(B\left)\right)=h\left(\rho \right(C\left)\right)=q$. We claim that, for any antichain D of P with $D\ne A$, $D\ne B$ and $D\ne C$, one has $h\left(\rho \right(D\left)\right)<q$. $D=A_1\cup B_1\cup C_1$ with $A_1\subset A\setminus B$, $B_1⊊B$, and $C_1⊊C\setminus B$. Now, we define two subsets of B:

$$\begin{array}{cc}\hfill B_2& =\{x_j\in B|x_j<x_i,x_i\in A_1\};\hfill \\ \hfill B_3& =\{x_j\in B|x_k<x_j,x_k\in C_1\}.\hfill \end{array}$$

Then $B_1\cap B_2=B_1\cap B_3=B_2\cap B_3=\varnothing$ and $B_1\cup B_2\cup B_3\subset B_3$. Let ${\sum }_{x_i\in A}c\left(i\right)=q_1$, ${\sum }_{x_j\in B_1}c\left(j\right)=q_2$, ${\sum }_{x_k\in C_1}c\left(k\right)=q_3$, ${\sum }_{x_j\in B_2}c\left(j\right)=q_1^{\prime }$, and ${\sum }_{x_j\in B_3}c\left(j\right)=q_3^{\prime }$. Since $A▵B$, $B▵C$ are connected, it follows that $q_1<q_1^{\prime }$ and $q_3<q_3^{\prime }$. Hence

$$\begin{array}{cc}\hfill h\left(\rho \right(D\left)\right)& =\sum _{x_i\in A_1}c\left(i\right)+\sum _{x_j\in B_1}c\left(j\right)+\sum _{x_k\in C_1}c\left(k\right)\hfill \\ \hfill & =q_1+q_2+q_3<q_1^{\prime }+q_2+q_3^{\prime }\hfill \\ \hfill & =\sum _{x_j\in B_2}c\left(j\right)+\sum _{x_j\in B_1}c\left(j\right)+\sum _{x_j\in B_3}c\left(j\right)\le \sum _{x_j\in B}c\left(j\right)=q.\hfill \end{array}$$

Thus $h\left(\rho \right(D\left)\right)<q$.

Case 4.$B\not\subset A$, $B\cap A\ne \varnothing$, $C\cap B=\varnothing$, and $C\cap A=\varnothing$.

Since P has rank 2, we can show $h\left(\rho \right(D\left)\right)<q$ by the same argument in Case 3 (Suppose $C\cap B=\varnothing$).

Case 5.$B\not\subset A$, $B\cap A=\varnothing$, $C\cap B=\varnothing$ and $C\cap A=\varnothing$.

Since P has rank 2, we can show $h\left(\rho \right(D\left)\right)<q$ by the same argument in Case 3 (Suppose $B\cap A=C\cap B=\varnothing$).

In conclusion, each $\mathcal{H}$ is a supporting hyperplane of $\mathcal{C}\left(P\right)$ and $\mathcal{H}\cap \mathcal{C}\left(P\right)$ coincides with the convex hull of $\left\{\rho \right(A),\rho (B),\rho (C\left)\right\}$, as desired. □

Corollary 1.

Triangles in 1-skeleton of $\mathcal{O}\left(P\right)$ or $\mathcal{C}\left(P\right)$ are in one-to-one correspondence with faces of 2-dimensional simplex of each polytope.