2.2. The proof of Lemma 2.1 [03D2]

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2.2. The proof of Lemma 2.1

We will describe a coherent subdivisions of $\Delta^{\vee}_{\lambda}$ (alternatively ${\Delta_{\nu}}$ ), which is isomorphic to $\Sigma$ . We hope that the method, which generalizes barycentric subdivisions, may find other applications in the future.

Definition.

Suppose that $P\subset Q\subset\mathbb{R}^{d}$ are polytopes such that $P$ is contained in the relative interior of some face of $Q$ . Then the result of pulling $P$ is by definition the coherent subdivision $\operatorname{pull}_{P}(Q)$ of $Q$ which is induced by the heights $1$ on $P$ , and $0$ on all faces of $Q$ that do not contain $P$ .

The case $P\subset\operatorname{relint}(Q)$ is used in [GP88] in order to prove that the space between $P$ and $Q$ can be triangulated without new vertices.

Figure 5: The pulling subdivision $\operatorname{pull}_{P}(Q)$ .

We can describe this subdivision combinatorially as follows. The faces of $\operatorname{pull}_{P}(Q)$ are all sets of the form $\operatorname{conv}(F\cup F^{\prime})$ for faces $F\preceq P$ and $F^{\prime}\prec Q$ with $F\not\subset F^{\prime}$ (including $F^{\prime}=\emptyset$ ) whose normal cones intersect in their relative interiors. That is, there should be a normal vector $n\in(\mathbb{R}^{d})^{*}$ which is maximized over $P$ precisely on $F$ , and over $Q$ precisely on $F^{\prime}$ . More generally, if $P$ is in the relative interior of a face of a polyhedral complex, then pulling $P$ will affect all faces that contain $P$ in the manner outlined above. If the original subdivision was coherent, then the pulling subdivision will remain coherent. This procedure generalizes the well studied pulling subdivisions where $P$ is a point [Grü67, § 5.2] or [Lee97].

We use these pullings in order to construct a generalized barycentric subdivisions below. We start with a purely combinatorial definition. For a poset $\mathcal{Q}$ , the poset/simplicial complex of chains in $\mathcal{Q}$ is denoted $\operatorname{bsd}(\mathcal{Q})$ .

Definition.

Suppose $\kappa\colon\mathcal{Q}\rightarrow\mathcal{P}$ is an order preserving, non-rank-increasing correspondence between the graded posets $\mathcal{Q}$ and $\mathcal{P}$ . Define the barycentric subdivision $\operatorname{bsd}(\mathcal{Q},\kappa)$ of $\mathcal{Q}$ with respect to $\kappa$ as the subposet

\{(p\ ,\ q_{0}\prec q_{1}\prec\ldots\prec q_{r})\in\mathcal{P}\times\operatorname{bsd}(\mathcal{Q})\ :\ p\preceq\kappa(q_{0})\}

of the product poset $\operatorname{bsd}(\mathcal{Q})\times\mathcal{P}$ .

If $\mathcal{P}$ has only one element, this specializes to $\operatorname{bsd}(\mathcal{Q})$ . In our applications, $\kappa$ will be clear from the context, and will write $\operatorname{bsd}(\mathcal{Q},\mathcal{P})$ instead.

One example of such a $\kappa$ arises in the following situation. Say that a polytope $P\subset\mathbb{R}^{d}$ is a Minkowski summand of the polytope $Q\subset\mathbb{R}^{d}$ , if there is an $\varepsilon>0$ , and a polytope $P^{\prime}\subset\mathbb{R}^{d}$ such that $\varepsilon P+P^{\prime}=Q$ . This is true if and only if the normal fan of $Q$ refines the normal fan of $P$ [Smi87]. So there is an order preserving, non-rank-decreasing correspondence on the level of the normal fans which turns into $\kappa\colon\mathcal{L}(Q)\rightarrow\mathcal{L}(P)$ on the level of the face lattices.

Definition.

Suppose that $P$ is a Minkowski summand of $Q$ . Define the barycentric subdivision $\operatorname{bsd}(Q;P)$ of $Q$ with respect to $P$ as follows. Start with $F=Q$ , and proceed by decreasing dimension of faces $F\preceq Q$ . Pull the translate $\varepsilon\kappa(F)-\varepsilon\widehat{\kappa(F)}+\widehat{F}$ of the corresponding face of $\varepsilon P$ in the relative interior of $F$ .

The usual barycentric subdivision appears as the special case where $P$ is a point.

Combinatorially, the face lattice $\mathcal{L}(\operatorname{bsd}(Q;P))$ is given by $\operatorname{bsd}(\mathcal{L}(Q),\mathcal{L}(P))$ . An element $(F_{0}\prec F_{1}\prec\cdots\prec F_{r},G)$ corresponds to the convex hull of the copies of $G\preceq P$ within the $F_{i}$ ’s:

\operatorname{conv}\left(\widehat{F}_{i}+\varepsilon(G-\widehat{\kappa(F_{i})})\right)=\operatorname{conv}\left(\widehat{F}_{i}-\varepsilon\widehat{\kappa(F_{i})}\right)+\varepsilon G

This polytope is the image under an affine embedding of the product of $G$ with an $r$ -simplex.

The notion generalizes to the situation of two polyhedral complexes with a realized order preserving, non-rank-increasing correspondence between their face posets. We will use this in Section 3.3 for the identity correspondence of $\mathcal{L}(T)$ .

Lemma 2.3.

There is a coherent subdivision of $\partial\Delta^{\vee}_{\lambda}$ which is combinatorially isomorphic to the restriction to $|\Sigma|$ of the product subdivision $\operatorname{bsd}(S)\times T$ .

Proof.

Here, $\Delta^{\vee}$ is a Minkowski summand of $\Delta^{\vee}_{\lambda}$ . Let $\psi$ be a piecewise linear concave function with domains of linearity given by $\operatorname{bsd}(\Delta^{\vee}_{\lambda};\Delta^{\vee})$ . Now $\nu$ induces another piecewise linear (non-concave) function $\tilde{\nu}$ on $\Delta^{\vee}_{\lambda}$ , which is $\nu$ in $G$ -direction on $(F_{0}\prec F_{2}\prec\cdots\prec F_{r},G)$ , and constant in $F_{i}$ -direction.

Figure 7: The function $\tilde{\nu}$ on $\operatorname{conv}\left[\begin{smallmatrix}-2&\ 1&\ 1&\ 2&\ 2\\ \ 0&\ 3&-3&\ 2&-2\\ -1&-1&-1&-1&-1\end{smallmatrix}\right]\prec\Delta^{\vee}_{\lambda}$ .

The function $\psi$ is strictly concave wherever $\tilde{\nu}$ is non-concave, so that for large $N$ , the function $N\psi+\tilde{\nu}$ will be concave. Its domains of linearity are products of simplices that correspond to simplices of $\operatorname{bsd}(\Delta^{\vee}_{\lambda})\cong\operatorname{bsd}(S)$ times simplices of $T$ . ∎

Now Lemma 2.1 follows if we start from $S$ and $\operatorname{bsd}(T)$ in stead of $S$ and $T$ .