2.1 Cones and fans

Let V be a finite dimensional real vector space of dimension n, and let $V^*$ denote its dual. Recall that a (closed convex) polyhedral cone in V is a set of the form

$$\begin{align*}\sigma = \operatorname{Cone}(W) = \left\{ \sum\limits_{w \in W} a_w w : a_w \geqslant 0 \right\} \subseteq V \end{align*}$$

with W a finite subset of V. Equivalently, there is a finite subset B of $V^*$ such that

$$\begin{align*}\sigma = \bigcap\limits_{b \in B} \left\{x \in V : b(x) \geqslant 0 \right\}. \end{align*}$$

We say that $\sigma $ is generated by W. Also, we write $\operatorname {Cone}(\emptyset ) = \{0\}$ . The dimension of $\sigma $ is the dimension of its linear span. The dual cone $\sigma ^\vee $ is defined as

$$\begin{align*}\sigma^\vee := \left\{ y \in V^* : y(x) \geqslant 0 \mbox{ for all } x \in \sigma\right\}. \end{align*}$$

Dual cones enjoy the property that if $\sigma $ is a polyhedral cone in V, then $\sigma ^\vee $ is a polyhedral cone in $V^*$ and $\sigma ^{\vee \vee } = \sigma $ .

For a face $\tau $ of $\sigma $ (denoted $\tau \preceq \sigma $ ), define its dual face

$$ \begin{align*} \tau^* &= \left\{y \in \sigma^\vee : y(x) = 0 \mbox{ for all } x\in\tau \right\} \\ &= \sigma^\vee \cap \tau^\perp. \end{align*} $$

Then $\tau ^*$ is a face of $\sigma ^\vee $ , $\tau ^{**} = \tau $ , $\tau \leftrightarrow \tau ^*$ is an inclusion-reversing bijection between faces of $\sigma $ and those of $\sigma ^\vee $ , and $\dim \tau + \dim \tau ^* = n$ . One-dimensional cones, i.e., half-lines, are called rays. A face $\tau $ of $\sigma $ is called a facet if $\dim \tau = \dim \sigma - 1$ , and its linear span is referred to as a wall of $\sigma $ . An edge is a face of dimension 1.

Define the relative interior $\sigma ^\circ $ of $\sigma $ to be the interior of $\sigma $ in its span. One then checks that $x \in \sigma ^\circ $ if and only if $y(x)> 0$ for all $y \in \sigma ^\vee \setminus \sigma ^\perp $ . A polyhedral cone $\sigma $ in V is strongly convex if the origin is a face. This is the case if and only if $\sigma $ contains no positive dimensional subspace of V if and only if $\sigma \cap (-\sigma ) = \{0\}$ if and only if $\dim \sigma ^\vee = n$ . A strongly convex polyhedral cone $\sigma \subseteq V$ is called simplicial if it is generated by linearly independent vectors. We note that the dual of a simplicial cone of maximal dimension is again simplicial.

For $y \in V^*$ , we set

$$\begin{align*}H_y := \left\{x \in V : y(x) = 0 \right\} \subseteq V \end{align*}$$

and define the closed (resp. open) spaces

$$\begin{align*}H^+_y := \left\{x \in V : y(x) \geqslant 0 \right\} \subseteq V \quad \mbox{ and } \quad H^-_y := \left\{x \in V : y(x) < 0 \right\} \subseteq V. \end{align*}$$

When $y \not = 0, \ H_y$ is a hyperplane and $H^+_y$ and $H^-_y$ are half-spaces in V. When $y=0$ , we have $H_y = H^+_y = V$ while $H_y^-$ is empty. If $\sigma \subseteq H^+_y$ for $y \not = 0$ , we say $H_y$ is a supporting hyperplane and $H^+_y$ (resp. $H^-_y$ ) is an inward (resp. outward) supporting half-space of $\sigma $ . (When $y=0$ , we automatically have $\sigma \subseteq H^+_0 = H_0 = V$ .) Note that $H_y$ is a supporting hyperplane of $\sigma $ if and only if $y \in \sigma ^\vee \setminus \{0\}$ . If $y_1, y_2, \dots , y_r$ generate $\sigma ^\vee ,$ then $\sigma = H^+_{y_1} \cap \cdots \cap H^+_{y_r}$ . Thus, every polyhedral cone is an intersection of finitely many closed half-spaces.

A fan $\Sigma $ in V is a finite collection of cones $\sigma \subseteq V$ satisfying the following three properties: (a) every $\sigma \in \Sigma $ is a strongly convex polyhedral cone, (b) for all $\sigma \in \Sigma ,$ each face of $\sigma $ also belongs to $\Sigma $ , and (c) for all $\sigma _1, \sigma _2 \in \Sigma $ , the intersection $\sigma _1 \cap \sigma _2$ is a face of each. The set of r-dimensional cones of $\Sigma $ is denoted by $\Sigma (r)$ . The support of $\Sigma $ is defined by

$$\begin{align*}|\Sigma| := \bigcup_{\sigma \in \Sigma} \, \sigma \subseteq V. \end{align*}$$

If $|\Sigma | = V$ , then $\Sigma $ is called a complete fan. A simplicial fan is a fan all whose cones are simplicial. Every fan can be refined into a simplicial fan.

Finally, for $\sigma \in \Sigma $ , we let $\Sigma / \sigma $ denote the fan in $V / \operatorname {Span}(\sigma )$ consisting of all the images of the cones $\sigma ' \succeq \sigma $ . If we fix an inner product on V, then $V / \operatorname {Span}(\sigma )$ can be identified with $\sigma ^\perp $ and $\Sigma / \sigma $ consists of projections of $\sigma ' \succeq \sigma $ onto $\sigma ^\perp $ .

2.2 Polytopes

A polytope is a set in $V^*$ of the form

$$\begin{align*}P = \operatorname{Conv}(S) = \left\{ \sum\limits_{u \in S} \lambda_u u : \lambda_u \geqslant 0, \sum_{u \in S} \lambda_u = 1 \right\}, \end{align*}$$

where S is a finite subset of $V^*$ . We say P is the convex hull of S. The dimension, $\dim P$ , of a polytope P is the dimension of the smallest affine subspace of $V^*$ containing P. Given $x \in V \setminus \{0\}$ and $r \in {\mathbb R}$ , we have the affine hyperplane

$$\begin{align*}H_{x,r} := \left\{ y \in V^* : y(x) = r \right\} \end{align*}$$

and the closed (resp. open) half-spaces

$$\begin{align*}H_{x,r}^{+} := \left\{ y \in V^* : y(x) \geqslant r \right\} \quad \mbox{ and } \quad H_{x,r}^{-} := \left\{ y \in V^* : y(x) < r \right\}. \end{align*}$$

A subset $Q \subseteq P$ is a face of P, denoted by $Q \preceq P$ , if there is $x \in V \setminus \{0\}$ and there is $r \in {\mathbb R}$ with

$$\begin{align*}Q = H_{x,r} \cap P \quad\mbox{ and }\quad P \subseteq H_{x,r}^+. \end{align*}$$

We then say that $H_{x,r}$ is a supporting affine hyperplane. The polytope P is regarded as a face of itself and faces of P of dimensions $0$ , $1$ , and $(\dim P - 1)$ are called vertices, edges, and facets, respectively.

A polytope $P \subseteq V^*$ can be written as a finite intersection of closed half-spaces, and an intersection

$$\begin{align*}P = \bigcap\limits_{i=1}^s H_{x_i,r_i}^+ \end{align*}$$

is a polytope provided that it is bounded. In general, an intersection of finitely many closed half-spaces is called a polyhedron and could be unbounded. When $\dim P = \dim V^*$ (i.e., full dimensional polytope) for each facet F, we have a unique supporting affine hyperplane and the corresponding closed half-space given by

$$\begin{align*}H_F = H_{u_F^+,a_F} = \left\{ y \in V^* : y(u^+_F) = a_F \right\} \end{align*}$$

and

$$\begin{align*}H_F^{+} = H^+_{u_F^+,a_F} = \left\{ y \in V^* : y(u^+_F) \geqslant a_F \right\}, \end{align*}$$

where $(u^+_F, a_F) \in V \times {\mathbb R}$ is unique up to multiplication by a positive real number. We call $u^+_F$ an inward-pointing facet normal of the facet F. Hence,

(2.1) $$ \begin{align} P = \bigcap\limits_{F \text{ facet }} H_F^+ = \left\{ y \in V^* : y(u^+_F) \geqslant a_F \mbox{ for all proper facets } F \prec P \right\}. \end{align} $$

This is the so-called facet representation of P. We also have a similar representation with outward-pointing facet normals $u^-_F = - u^+_F$ . When the facet normals $u^{\pm }_F$ are assumed to be unit vectors, we may call the $a_F$ the support numbers of P.

Let Q be a face of P and define the inward (resp. outward) tangent cone $T^+_{P, Q}$ (resp. $T^-_{P, Q}$ ) via

(2.2) $$ \begin{align} \phantom{\text{resp.~\, }}T^+_{P, Q} &:= \left\{ y \in V^* : y(u^+_F) \geqslant a_F \mbox{ for all facets } F \supset Q \right\}, \end{align} $$

(2.3) $$ \begin{align} \text{resp. } T^-_{P, Q}&:=\left\{ y \in V^* : y(u^+_F) < a_F \mbox{ for all facets } F \supset Q \right\} \end{align} $$

$$\begin{align*}&\phantom{\text{resp.~\, QP~~~}}= \left\{ y \in V^* : y(u^-_F)> a_F \mbox{ for all facets } F \supset Q \right\}. \notag \end{align*}$$

See Figures 3 and 4 for illustrations of inward and outward tangent cones of a quadrilateral at a vertex and at an edge, respectively.

A polytope $P \subseteq V^*$ of dimension d is called a d-simplex (or just a simplex) if it has $d+1$ vertices, simplicial if every facet is a simplex, and simple if every vertex is the intersection of precisely d facets.

Given a polytope $P = \operatorname {Conv}(S)$ , its multiple $rP = \operatorname {Conv}(rS)$ is also a polytope for any $r \geqslant 0$ . The Minkowski sum $P_1 + P_2 = \{y_1+y_2 : y_i \in P_i\}$ of two polytopes $P_1 = \operatorname {Conv}(S_1)$ and $P_2 = \operatorname {Conv}(S_2)$ is again a polytope, and we have the distributive law $rP + sP = (r+s) P$ . The set $\mathcal {P}(V^*)$ of polytopes in $V^*$ together with the Minkowski sum is a cancellative semigroup. The following theorem is originally due to Minkowski.

There is also a discrete analogue of Theorem 2.1 which is harder and more subtle to prove. It is a generalization of the notion of the Ehrhart polynomial. Let $M \cong {\mathbb Z}^n$ be a full rank lattice in $V^* \cong {\mathbb R}^n$ . Let ${\mathcal P}(M)$ denote the collection of lattice polytopes with respect to M, that is, all polytopes in $V^*$ whose vertices belong to M. The set ${\mathcal P}(M)$ is closed under the Minkowski sum and multiplication by positive integers.

More generally, the polynomiality property holds for any valuation (also called finitely additive measure). A function $\Phi : {\mathcal P}(M) \to {\mathbb R}_{\geqslant 0}$ is called a valuation if for all $P_1, P_2 \in {\mathcal P}(M)$ , the following hold:

1. (1) $\Phi $ is monotone with respect to inclusion, i.e., $\Phi (P_1) \leq \Phi (P_2)$ provided that $P_1 \subset P_2$ .

2. (2) $\Phi (P_1 \cup P_2) = \Phi (P_1) + \Phi (P_2) - \Phi (P_1 \cap P_2)$ .

We say $\Phi $ is ${{\mathbb Z}}^n$ -invariant if $\Phi (m+P) = \Phi (P)$ for all $P \in {\mathcal P}(M)$ and $m \in M$ . The following is a beautiful result of McMullen [Reference McMullenMc77]. It generalizes Theorem 2.2.

2.3 Normal fan

For $Q \preceq P$ , let

$$\begin{align*}\sigma_Q := \operatorname{Cone}\left( u^-_F : \text{ facets } F \supset Q \right). \end{align*}$$

Given a full dimensional polytope $P \subseteq V^*$ , the cones $\sigma _Q$ fit together to form the normal fan of P in V given by

$$\begin{align*}\Sigma_P = \left\{ \sigma_Q : Q \preceq P \right\}. \end{align*}$$

Note that we have used outward facet normals $u^-_F$ to define the normal fan (see Figure 5). (Some authors use inward facet normals $u^+_F$ instead.)

Figure 5: A polygon and its normal fan. Note that in our convention, we use outward facet normals to define the cones in the normal fan.

Let $\mathcal {P}(\Sigma )$ be the collection of all convex polytopes whose normal fan is $\Sigma $ . This set is closed under the Minkowski sum of polytopes and multiplication by positive scalars. For $P \in \mathcal {P}(\Sigma )$ , we have an inclusion-reversing bijection

(2.4) $$ \begin{align} Q = Q_\sigma \longleftrightarrow \sigma = \sigma_Q \end{align} $$

between the set of faces of P and the set of cones in the normal fan $\Sigma $ . In particular, the facets F of P correspond to rays $\rho \in \Sigma (1)$ . For a ray $\rho \in \Sigma (1)$ , we set $a_\rho = a_F$ , where F is the facet corresponding to $\rho $ and $a_F$ are the support numbers of P (see (2.1)). The map $P \mapsto (a_\rho )_{\rho \in \Sigma (1)}$ gives an embedding of $\mathcal {P}(\Sigma )$ into ${\mathbb R}^s$ , where $s=|\Sigma (1)|$ . The image is a full dimensional (open) convex polyhedral cone.

Let P be a full dimensional polytope with normal fan $\Sigma _P$ . Let $Q \preceq P$ be a face with corresponding cone $\sigma _Q \in \Sigma _P$ . Then the normal fan $\Sigma _Q$ (of the polytope Q) is the fan $\Sigma _P / \sigma _Q$ (defined at the end of Section 2.1). It consists of the images of the cones $\sigma ' \succeq \sigma _Q$ in the quotient vector space $V / \operatorname {Span}(\sigma _Q)$ .

2.4 Nearest face partition

Fix an inner product $\langle \cdot , \cdot \rangle $ on V. Let $P \subset V$ be a convex polyhedron. To P, we can associate a partition of V into polyhedral regions $V_P^Q$ as follows. For each face $Q \preceq P$ , let

$$ \begin{align*}V_P^Q &= \left\{ x \in V : \text{the minimum distance from } x \text{ to } P \text{ is attained}\right. \\&\qquad \left.\text{at a point in the relative interior of } Q \right\}.\end{align*} $$

The following is straightforward to verify.

We can modify the $V_P^Q$ to obtain a slightly different partition $\left \{W_P^Q : Q \preceq P \right \}$ . For each face $Q \preceq P$ , let

$$ \begin{align*}W_P^Q = \overline{V_P^Q} \setminus \bigg(\bigcup_{Q' \gneqq Q} \overline{V_P^{Q'}}\bigg),\end{align*} $$

where $\overline {V_P^Q}$ denotes the closure of $V_P^Q$ . The polyhedra $W_P^Q$ and $V_P^Q$ have the same relative interior, but they are different on the boundary.

We refer to both $\left \{V_P^Q : Q \preceq P \right \}$ and $\left \{W_P^Q : Q \preceq P \right \}$ as the nearest face partition of V with respect to the polyhedron P (see Figure 6). We note that if, in particular, $P = \sigma $ is a cone (with apex at the origin), then the closure of the parts in the partition with respect to $\sigma $ in fact form a complete fan in V. In practice, we will also use the nearest face partition to partition a polyhedron inside V.

Figure 6: Nearest face partition for a polygon illustrating polyhedral regions $V_P^Q$ and $W_P^Q$ corresponding to an edge Q.

2.5 Conical decomposition theorems

We end this section by recalling two beautiful formulas which represent the characteristic function of a polytope as an alternating sum of characteristic functions of cones. For a nice overview of these decompositions and related topics, we refer the reader to [Reference Beck, Haase and SottileBHS09].

2.5.1 Brianchon–Gram theorem

The first conical decomposition theorem we discuss is the Brianchon–Gram theorem. It is named after Brianchon and Gram who independently proved the $n = 3$ case in 1837 and 1874, respectively ([Reference BrianchonB37, Reference GramG1874]). It is the mother of all cone decompositions! See [Reference HaaseHass05, Section 1.1] and the references therein. Also, see [Reference AgapitoAg06].

Theorem 2.6 (Brianchon–Gram)

Let P be a polytope in $V^*$ . We have the following equality, where ${\textbf 1}$ denotes characteristic function:

(2.5) $$ \begin{align} {{\textbf 1}}_P = \sum_{Q \preceq P} (-1)^{\dim Q} {{\textbf 1}}_{T^+_{P, Q}}. \end{align} $$

Proof For a point $y \in P$ , the right-hand side computes the Euler characteristic of P and hence is equal to $1$ since P is contractible. For $y \notin P$ , we have to subtract the Euler characteristic of the subcomplex that is visible from y which is again contractible.▪

Alternatively, one can formulate Brianchon–Gram in terms of outward-looking tangent cones.

Theorem 2.7 (Brianchon–Gram, alternative version)

Let P be a polytope in $V^*$ . We have the following equality:

(2.6) $$ \begin{align} {{\textbf 1}}_{P} = \sum_{Q \preceq P} (-1)^{n - \dim Q} {{\textbf 1}}_{T^-_{P, Q}}. \end{align} $$

The above version of the Brianchon–Gram formula looks similar to Arthur’s definition of the modified kernel $k^T(x)$ , as was observed in [Reference CasselmanCass04]. See Figures 7 and 8 for illustrations of (2.5) and (2.6).

Figure 7: Illustration of the Brianchon–Gram theorem (inward-looking tangent cones) for a triangle.

Figure 8: Illustration of the Brianchon–Gram theorem (alternative version, outward-looking tangent cones) for a triangle.

2.5.2 Lawrence–Varchenko theorem

The second conical decomposition due to Lawrence [Reference LawrenceLaw91] and Varchenko [Reference VarchenkoVr87] represents the characteristic function of a polytope as an alternating sum of characteristic functions of certain cones associated with vertices of the polytope. It is a predecessor to the work of Khovanskii and Pukhlikov [Reference Khovanskii and PukhlikovKP93a, Reference Khovanskii and PukhlikovKP93b] and Brion and Vergne [Reference BrionBr88, Reference Brion and VergneBV97]. It is related to Morse theory on polytopes as well as equivariant cohomology of toric varieties. The Lawrence–Varchenko theorem follows immediately from Khovanskii–Pukhlikov results as well (see [Reference Khovanskii and PukhlikovKP93b, Section 3.2]).

Let $P \subset V$ be a simple polytope, and let v be a vertex of P. Let $w_1, \ldots w_r$ be edge vectors of P at the vertex v. Fix a dual vector $\xi \in V^*$ such that $\langle w_i, \xi \rangle \neq 0$ , for all i. We define vectors $w^{\prime }_1, \ldots , w^{\prime }_r$ as follows:

$$ \begin{align*} w^{\prime}_i = \begin{cases} w_i, & \mbox{ if } \langle w_i, \xi \rangle> 0, \\ - w_i, & \mbox{ otherwise. } \end{cases} \end{align*} $$

Finally, define the polarized tangent cone $T^\xi _{P, v}$ with apex at v by

$$\begin{align*}T^\xi_{P, v} = \left\{ \sum_{i=1}^r \lambda_i w^{\prime}_i : \begin{array}{ll} \lambda_i \geqslant 0 \text{ if } w^{\prime}_i = w_i \\ \lambda_i> 0 \text{ if } w^{\prime}_i = -w_i \end{array} \right\}. \end{align*}$$

Theorem 2.8 (Lawrence–Varchenko)

With notation as above, we have the following:

(2.7) $$ \begin{align} {\textbf 1}_P = \sum_{v} (-1)^{n_v} {\textbf 1}_{T^\xi_{P, v}}, \end{align} $$

where the sum is over all the vertices v of P, and $n_v = |\{i : w^{\prime }_i = -w_i \}|$ .

See Figure 9 for an illustration of (2.7).

Figure 9: Illustration of the Lawrence–Varchenko theorem for a quadrangle.

2.6 Khovanskii–Pukhlikov virtual polytopes and convex chains

This is a summary of some ideas and results from [Reference Khovanskii and PukhlikovKP93a, Reference Khovanskii and PukhlikovKP93b] that we will need later. As before, $V \cong {\mathbb R}^n$ denotes an n-dimensional real vector space.

Recall that ${\mathcal P}(V^*)$ denotes the set of polytopes in the dual space $V^*$ . The set $\mathcal {P}(V^*)$ is equipped with the operations of Minkowski sum and multiplication by positive scalars. One knows that ${\mathcal P}(V^*)$ together with the Minkowski sum is a cancellative semigroup and hence it can be extended to a real vector space $\mathcal {V}(V^*)$ consisting of formal differences $P_1 - P_2$ , $P_i \in \mathcal {P}(V^*)$ , where for polytopes $P_1, P_2, P^{\prime }_1, P^{\prime }_2$ , we have $P_1 - P_2 = P^{\prime }_1 - P^{\prime }_2$ if and only if $P_1 + P^{\prime }_2 = P^{\prime }_1 + P_2$ .

We note that ${\mathcal V}(V^*)$ is an infinite dimensional vector space.

Let $\Sigma $ be a complete fan in V. Recall that ${\mathcal P}(\Sigma )$ denotes the set of all polytopes in $V^*$ whose normal fan is $\Sigma $ . The set ${\mathcal P}(\Sigma )$ is closed under the Minkowski sum and multiplication by positive scalars. We denote by $\mathcal {V}(\Sigma )$ the subspace of $\mathcal {V}(V^*)$ spanned by $\mathcal {P}(\Sigma )$ . The elements of $\mathcal {V}(\Sigma )$ are called virtual polytopes with normal fan $\Sigma $ . Generalizing the facet representation of a polytope $P \in {\mathcal P}(\Sigma )$ , i.e., representation as an intersection of half-spaces $H^+_{u^+_\rho , a_\rho }$ , $\rho \in \Sigma (1)$ , each virtual polytope in ${\mathcal V}(\Sigma )$ is represented by a collection of oriented hyperplanes $H_{u_\rho , a_\rho }$ , $\rho \in \Sigma (1)$ . Note that any choice of the support numbers $a_\rho $ yields a virtual polytope (even if the intersection of the corresponding half-spaces is empty). See Figures 10 and 11 for illustrations of a usual and virtual quadrangle with the same normal fan.

Figure 10: A usual quadrangle with its normal fan.

Figure 11: A virtual quadrangle with the same normal fan.

Each polytope $P \in {\mathcal P}(V^*)$ is determined by its characteristic function ${\textbf 1}_P: V^* \to \{0, 1\}$ . We would like to extend the assignment $P \mapsto {\textbf 1}_P$ to virtual polytopes. The natural extension of the set of characteristic functions of convex polytopes (to a vector space) is the set of convex chains (defined by Khovanskii and Pukhlikov).

Moreover, in general, one can consider the characteristic functions of convex polyhedral cones.

A remarkable construction in [Reference Khovanskii and PukhlikovKP93a] is a “convolution” operation $*$ on $\mathcal {Z}(V^*)$ which makes it a commutative algebra (together with addition and scalar multiplication of functions). It has the property that for any two polytopes $P_1$ and $P_2$ , we have

$$\begin{align*}{\textbf 1}_{P_1} * {\textbf 1}_{P_2} = {\textbf 1}_{P_1 + P_2}.\\[-15pt] \end{align*}$$

In particular, the identity element for the $*$ operation is ${\textbf 1}_{\{0\}}$ , the characteristic function of the origin.

For a polytope P, it is shown in [Reference Khovanskii and PukhlikovKP93a] that the inverse (with respect to $*$ ) of ${\textbf 1}_P$ is the convex chain $(-1)^{\dim P} {\textbf 1}_{P^\circ }$ , where $P^\circ $ denotes the relative interior of P. In other words,

$$\begin{align*}{\textbf 1}_P * (-1)^{\dim P} {\textbf 1}_{P^\circ} = {\textbf 1}_{\{0\}}. \end{align*}$$

One verifies that

$$\begin{align*}(-1)^{\dim P} {\textbf 1}_{P^\circ} = \sum_{Q \preceq P} (-1)^{\dim Q} {\textbf 1}_Q, \end{align*}$$

and hence $(-1)^{\dim P} {\textbf 1}_{P^\circ }$ is indeed a convex chain. It follows that

(2.8) $$ \begin{align} \iota: P_1 - P_2 \mapsto {\textbf 1}_{P_1} * (-1)^{\dim P_2} {\textbf 1}_{P_2^\circ} = \sum_{Q \preceq P_2} (-1)^{\dim Q} {\textbf 1}_{P_1 + Q} \end{align} $$

defines a natural embedding of the group of virtual polytopes (with the Minkowski sum) into the semigroup of convex chains (with convolution $*$ ). We refer to the right-hand side of (2.8) as the convex chain associated with the characteristic function of the virtual polytope $P_1 - P_2$ . In fact, it is shown in [Reference Khovanskii and PukhlikovKP93a] that the image of $\iota $ coincides with the set of $*$ -invertible convex chains.

We can talk about vertices of a virtual polytope. For a virtual polytope $P \in {\mathcal V}(\Sigma )$ , the vertices are in one-to-one correspondence with the full dimensional cones in $\Sigma $ . Similarly, the notion of a tangent cone of a polytope extends to virtual polytopes. The tangent cones of $P \in {\mathcal V}(\Sigma )$ are in one-to-one correspondence with $\sigma \in \Sigma $ .

There is a generalization of the Brianchon–Gram theorem to convex chains (see [Reference Khovanskii and PukhlikovKP93a, Section 4, Proposition 2]). The Lawrence–Varchenko theorem also extends to simple virtual polytopes.

Theorem 2.10 (Lawrence–Varchenko for virtual polytopes)

Let P be a virtual polytope in $V^*$ , and let $\pi : V^* \to {\mathbb R}$ be the corresponding convex chain. Then

(2.9) $$ \begin{align} \pi = \sum_{v} (-1)^{n_v} {\textbf 1}_{T^\xi_{P, v}}, \end{align} $$

where the sum is over all the vertices v of P and $T^\xi _{P, v}$ and $n_v$ are as in Theorem 2.8.

See Figure 12 for an illustration of (2.9).

Figure 12: Illustration of the Lawrence–Varchenko theorem for a virtual quadrangle.

2.7 Incidence algebra of a poset and Möbius inversion

For a nice reference about incidence algebra and Möbius inversion, see [Reference StanleySt12, Sections 3.6 and 3.7]. Let $\mathcal {P}$ be a finite poset with partial order $\prec $ . Let R be a commutative ring with $1$ which we take as the ring of scalars. Let $\tilde {\mathcal {P}} = \{(\tau , \sigma ) : \tau \preceq \sigma \} \subset \mathcal {P} \times \mathcal {P}$ be the collection of all intervals in $\mathcal {P}$ . Let $I(\mathcal {P}) = \{F: \tilde {\mathcal {P}} \to R \}$ be the set of functions from $\tilde {\mathcal {P}}$ to R. Clearly, $I = I(\mathcal {P})$ is an abelian group with addition of functions. One defines a convolution operation $*$ on I as follows. For $F, G \in I$ define $F * G \in I$ by

$$\begin{align*}(F * G)(\tau, \sigma) = \sum_{\tau \preceq \tau' \preceq \sigma} F(\tau, \tau') G(\tau', \sigma). \end{align*}$$

It can be verified that $(I, +, *)$ is an algebra over R, called incidence algebra of the poset $\mathcal {P}$ . In general, $I(\mathcal {P})$ is not commutative.

The identity (for the convolution operation $*$ ) is the function $\delta $ defined by

$$\begin{align*}\delta(\tau, \sigma) = \begin{cases} 1, \quad \tau = \sigma, \\ 0, \quad \tau \neq \sigma. \end{cases} \end{align*}$$

A distinguished element of the incidence algebra is the constant function $\zeta (\tau , \sigma ) = 1$ , for any interval $\tau \preceq \sigma $ . The Möbius inversion formula states that the function $\zeta $ is invertible and its inverse is the Möbius function $\mu $ . For the general poset $\mathcal {P}$ , the Möbius function is constructed/defined inductively, but in specific examples, it can be defined/computed explicitly.

Example 2.11 (Poset of subsets of a finite set)

Let $\mathcal {P}$ be the poset of all subset of $\{1, \ldots , d\}$ ordered by inclusion. It can be shown that the Möbius function in this case is given by

$$\begin{align*}\mu(I, J) = (-1)^{|I| - |J|}, \quad J \subset I, \end{align*}$$

and the Möbius inversion formula recovers the inclusion–exclusion principle.

The following is the main example of a poset that we will be concerned with in the paper.

Example 2.12 (Poset of faces of a convex polyhedral cone)

Let $\mathcal {P}$ be the poset of all faces of a given convex polyhedral cone $C \subset {\mathbb R}^n$ . If $\sigma $ is simplicial of dimension d, then this poset is the same as the poset of all subsets of $\{1, \ldots , d\}$ above. It can be shown that the Möbius function in this case is given by

$$ \begin{align*}\mu(\tau, \sigma) = (-1)^{\dim \sigma - \dim \tau}, \quad \tau \preceq \sigma.\end{align*} $$

4.1 An extension of the Langlands combinatorial lemma

As before, V is an n-dimensional real vector space. We fix an inner product $\langle \cdot , \cdot \rangle $ on V and identify V with its dual space $V^*$ . Let $\Sigma $ be a full dimensional, complete, simplicial fan in V, and let $\Delta \in \mathcal {P}(\Sigma )$ be a full dimensional simple polytope with normal fan $\Sigma $ . Since we identified V and $V^*$ , we take both $\Sigma $ and $\Delta $ to lie in V.

Let $\sigma \in \Sigma $ be a cone. First, we consider the case where $\sigma $ is full dimensional. Let $v_\sigma $ be the corresponding vertex of $\Delta $ . Let $W = \{w_1, \ldots , w_n\}$ (resp. $B = \{b_1, \ldots , b_n\}$ ) be the set of edge vectors of $\sigma $ (resp. of $\sigma ^\vee $ ). Then the $b_i$ (resp. the $w_j$ ) are the inward facet normals to $\sigma $ (resp. $\sigma ^\vee $ ), and the cone $\sigma $ is given by inequalities as

$$\begin{align*}\sigma = \left\{ x : \langle x, b_i \rangle \geqslant0,~i=1, \ldots, n \right\}. \end{align*}$$

Also, the inward-looking tangent cone $T^+_{\Delta , \sigma }$ at the vertex $v_\sigma $ is given by

$$\begin{align*}T^+_{\Delta, \sigma} = \left\{ x : \langle x, w_i \rangle \leqslant \langle v_\sigma, w_i \rangle,~ i=1, \ldots, n \right\}. \end{align*}$$

We consider the oriented hyperplanes corresponding to the union of these two sets of inequalities:

(4.1) $$ \begin{align} H_{b_i, 0} &= \left\{x : \langle x, b_i \rangle = 0 \right\}, & i=1, \ldots, n, \nonumber \\ H_{w_i, \langle v_\sigma, w \rangle} &= \left\{x : \langle x, w_i \rangle = \langle v_\sigma, w \rangle \right\}, & i=1, \ldots, n. \end{align} $$

If $v_\sigma $ lies in $\sigma $ , then the hyperplanes in (4.1) are the facets of the polytope $\Delta \cap \sigma $ oriented outward. In general, $v_\sigma $ may not lie in $\sigma $ .

See Section 2.6 for a review of the notions of virtual polytope and convex chain. Also, see Figure 16 for a three-dimensional example of $\Gamma _{\Delta ,\sigma }$ and Figure 17 for a pair of two-dimensional examples of the virtual polytope $\Gamma _{\Delta , \sigma }$ and its convex chain $\gamma _{\Delta , \sigma }$ .

Figure 16: A three-dimensional example where $\Gamma _{\Delta , \sigma }$ is a cube. A face $\tau $ (of $\sigma $ ) and its corresponding dual face $\tau ^*$ (of $\sigma ^\vee $ ) and the vertex $v_\tau $ (of $\Gamma _{\Delta , \sigma }$ ) are illustrated.

Figure 17: Two examples of the virtual polytopes $\Gamma _{\Delta , \sigma }$ . In the first example, the vertex $v_\sigma $ lies in the cone $\sigma $ and $\Gamma _{\Delta , \sigma }$ is an actual polytope (a quadrangle). The convex chain $\gamma _{\Delta , \sigma }$ is the characteristic function of the quadrangle. In the second example, $v_\sigma $ lies outside $\sigma $ and $\Gamma _{\Delta , \sigma }$ is a virtual quadrangle. The convex chain $\gamma _{\Delta , \sigma }$ is the function which has values $1$ and $-1$ in the two shaded regions, respectively.

In this section, we consider the Lawrence–Varchenko conical decomposition for the virtual polytope $\Gamma _{\Delta , \sigma }$ (Theorem 2.10). We will see that this recovers and extends some of the key combinatorial lemmas appearing in Arthur’s work (e.g., [Reference ArthurAr81]). As a special case, we immediately recover the Langlands combinatorial lemma (see [Reference ArthurAr05, Section I.8, p. 46], [Reference Goresky, Kottwitz and MacPhersonGKM97, Appendix B]). In addition, we interpret the Langlands combinatorial lemma as a formula for the inverse of a distinguished element in the incidence algebra of poset of faces of $\sigma $ (see Section 2.7).

Recall that for $\tau \preceq \sigma $ , the largest face of $\sigma ^\vee $ orthogonal to $\tau $ is denoted by $\tau ^*$ and we have $\dim \tau + \dim \tau ^* = n$ (Section 2.1). It follows that the intersection $\operatorname {Span}(\tau ) \cap (v_\sigma + \operatorname {Span}(\tau ^*))$ is a single point which can be shown to be a vertex $v_\tau $ of $\Gamma _{\Delta , \sigma }$ . In fact, we will see below that $\tau \mapsto v_\tau $ gives a one-to-one correspondence between the faces of $\sigma $ and the vertices of $\Gamma _{\Delta , \sigma }$ . The vertex corresponding to the zero-dimensional face $0$ is $0$ itself. On the other hand, the vertex corresponding to the whole $\sigma $ is the vertex $v_\sigma $ of  $\Delta $ .

For a face $\tau \preceq \sigma $ , let $W(\tau ) \subset W$ (resp. $B(\tau ) \subset B$ ) be the subset of edge vectors of $\tau $ (resp. $\tau ^*$ ). Thus,

$$ \begin{align*} \tau = \{ x \in \sigma : \langle x, b \rangle = 0,~ b \in B(\tau) \}. \end{align*} $$

The vertex $v_\tau $ is then the unique solution of the system of equations

$$\begin{align*}\begin{cases} \langle x, w \rangle = \langle v_\sigma, w \rangle, & \forall w \in W(\tau), \\ \langle x, b \rangle = 0, & \forall b \in B(\tau). \\ \end{cases} \end{align*}$$

Moreover, the inward tangent cone $T^+_{\Gamma _{\Delta , \sigma }, v_\tau }$ at the vertex $v_\tau $ is given by the inequalities

$$\begin{align*}T^+_{\Gamma_{\Delta, \sigma}, v_\tau} = \left\{x \in V : \begin{array}{ll} \langle x, w \rangle \leqslant \langle v_\sigma, w \rangle, & \forall w \in W(\tau) \\ \langle x, b \rangle \geqslant0, & \forall b \in B(\tau) \end{array} \right\}. \end{align*}$$

Thus, the set of outward facet normals of $\Gamma _{\Delta , \sigma }$ at $v_\tau $ is $W(\tau ) \cup -B(\tau )$ . In other words, the cone in the normal fan of $\Gamma _{\Delta , \sigma }$ corresponding to the vertex $v_\tau $ is generated by the set of vectors $W(\tau ) \cup -B(\tau )$ (Section 2.3).

Consider the nearest face partition corresponding to $\sigma $ (Section 2.4). That is, for each face $\tau $ , let $V_\sigma ^\tau $ be the set of points $x \in V$ whose shortest distance to $\sigma $ is attained at a point in the relative interior of $\tau $ . Since $\sigma $ is a cone, each $V_\sigma ^\tau $ is a full dimensional cone. Moreover, the closures of the cones $V_\sigma ^\tau $ , $\tau \preceq \sigma $ , are the maximal cones of a complete simplicial fan in V which we call the nearest face fan of $\sigma $ . The following is straightforward to verify.

Since the $V_\sigma ^\tau $ partition of the whole space V, the above proposition shows that the union of the cones generated by $W(\tau ) \cup -B(\tau )$ , $\tau \preceq \sigma $ , is V. This then implies that the normal fan of $\Gamma _{\Delta , \sigma }$ coincides with the nearest fan of $\sigma $ . In particular, the $v_\tau $ are all of the vertices of $\Gamma _{\Delta , \sigma }$ . In other words, $\tau \mapsto v_\tau $ gives a one-to-one correspondence between the faces of $\sigma $ and the vertices of $\Gamma _{\Delta , \sigma }$ .

Now, take a vector $\xi $ in $\sigma ^\circ \cap (\sigma ^\vee )^\circ $ , that is,

$$ \begin{align*} \langle \xi, b \rangle> 0,~ \forall b \in B, \end{align*} $$

$$ \begin{align*} \langle \xi, w \rangle> 0,~ \forall w \in W. \end{align*} $$

Note that since $\sigma \neq V$ , we know $(\sigma ^\circ )^\vee + \sigma ^\circ \neq V$ and hence $\sigma ^\circ \cap (\sigma ^\vee )^\circ = ((\sigma ^\circ )^\vee + \sigma ^\circ )^\vee \neq \emptyset $ .

Let $T^\xi _{\Gamma _{\Delta , \sigma }, v_\tau }$ be the polarized tangent cone at the vertex $v_\tau $ appearing in the Lawrence–Varchenko decomposition of $\Gamma _{\Delta , \sigma }$ relative to the vector $\xi $ (see Section 2.5). By construction, the edge vectors of $T^\xi _{\Gamma _{\Delta , \sigma }, v_\tau }$ are $\pm $ the edge vectors of the tangent cone of $\Gamma _{\Delta , \sigma }$ at $v_\tau $ so that the minimum of $\langle \xi , \cdot \rangle $ on $T^\xi _{\Gamma _{\Delta , \sigma }, v_\tau }$ is attained at the vertex $v_\tau $ . Since the inner product of $\xi $ with any vector in $W \cup B$ is positive, it follows that the set of inward facet normals of $T^\xi _{\Gamma _{\Delta , \sigma }, v_\tau }$ is exactly $W(\tau ) \cup B(\tau )$ . More precisely, $T^\xi _{\Gamma _{\Delta , \sigma }, v_\tau }$ is defined by the inequalities

(4.2) $$ \begin{align} T^\xi_{\Gamma_{\Delta, \sigma}, v_\tau} = \left\{ x \in V : \begin{array}{ll} \langle x, w \rangle> \langle v_\sigma, w \rangle, & \forall w \in W(\tau) \\ \langle x, b \rangle \geqslant 0, & \forall b \in B(\tau) \end{array} \right\}. \end{align} $$

On the other hand, let $C_\sigma ^\tau $ be the inward-looking tangent cone of $\sigma $ at $\tau $ . It is the cone defined as

$$ \begin{align*} C_\sigma^\tau = \{ x \in V : \langle x, b \rangle \geqslant 0,~\forall b \in B(\tau)\}. \end{align*} $$

It follows from (4.2) that $T^\xi _{\Gamma _{\Delta , \sigma }, v_\tau }$ can be written as

$$ \begin{align*} T^\xi_{\Gamma_{\Delta, \sigma}, v_\tau} = C_\sigma^\tau \cap T^-_{\Delta, \tau}. \end{align*} $$

If $\sigma $ is not full dimensional, we can repeat the above, replacing $\Delta $ with $\Delta \cap \operatorname {Span}(\sigma )$ . Then $\gamma _{\Delta , \sigma }$ is a convex chain supported on $\operatorname {Span}(\sigma )$ . We extend $\gamma _{\Delta , \sigma }$ to the whole V by requiring it to be constant along $\sigma ^\perp $ . Now, applying the Lawrence–Varchenko theorem to the virtual polytope $\Gamma _{\Delta , \sigma }$ and the vector $\xi $ as above, we obtain the following conical decomposition for $\Gamma _{\Delta , \sigma }$ .

Lemma 4.6 With notation as above, let $\gamma _{\Delta , \sigma }$ be the convex chain associated with

the virtual polytope $\Gamma _{\Delta , \sigma }$ . We have

(4.3) $$ \begin{align} \gamma_{\Delta, \sigma} = \sum_{\tau \preceq \sigma} (-1)^{\dim \tau} {\textbf 1}_{C_\sigma^\tau} {\textbf 1}_{T^-_{\Delta, \tau}}. \end{align} $$

Letting $\Delta = \{0\}$ , we recover a combinatorial lemma of Langlands.

Corollary 4.7 (Langlands combinatorial lemma)

Let $\sigma \subset V$ be a convex polyhedral cone. The following identities hold.

(4.4) $$ \begin{align} \sum_{\tau \preceq \tau' \preceq \sigma} (-1)^{\dim \tau + \dim \tau'} {\textbf 1}_{C_\sigma^{\tau'}} {\textbf 1}_{C_{\tau^*}^{\tau^{\prime *}}} = \begin{cases} 1, & \mbox{ if } \tau = \sigma, \\ 0, & \mbox{ if } \tau \neq \sigma, \end{cases} \end{align} $$

(4.5) $$ \begin{align} \sum_{\tau \preceq \tau' \preceq \sigma} (-1)^{\dim \tau' + \dim \tau} {\textbf 1}_{C_\tau^{\tau'}} {\textbf 1}_{C_{\sigma^*}^{\tau^{\prime *}}} = \begin{cases} 1, & \mbox{ if } \tau = \sigma, \\ 0, & \mbox{ if } \tau \neq \sigma. \end{cases} \end{align} $$

Alternatively, consider the incidence algebra of the poset of faces of $\sigma $ with ring of scalars R being the ring of all real-valued functions on V (see Section 2.7 and Example 2.12 ). Define the elements F and G of the incidence algebra by

$$ \begin{align*} F(\tau, \tau') &= (-1)^{\dim \tau} {\textbf 1}_{C_\tau^{\tau'}},\\ G(\tau, \tau') &= (-1)^{\dim \tau} {\textbf 1}_{C_{\tau^*}^{\tau^{\prime *}}}. \end{align*} $$

Equations ( 4.4 ) and ( 4.5 ) state that F and G are inverses of each other in the incidence algebra, that is,

(4.6) $$ \begin{align} (F * G)(\tau, \sigma) = (G * F)(\tau, \sigma) = \delta(\tau, \sigma). \end{align} $$

Corollary 4.8 With notation as before, we have

(4.7) $$ \begin{align} {\textbf 1}_{T^-_{\Delta, \sigma}} = \sum_{\tau \preceq \sigma} (-1)^{\dim \tau} {\textbf 1}_{C_{\sigma}^{\tau}} \, \gamma_{\Delta, \tau}. \end{align} $$

4.2 Proof of polynomiality

Proof of Theorem 4.1

In the definition of $J_{\Sigma }(\Delta )$ , we use Corollary 4.8 to write $T^-_{\Delta , \sigma }$ as $\sum \limits _{\tau \preceq \sigma } (-1)^{\dim \tau } {\textbf 1}_{C_{\sigma }^{\tau }} \, \gamma _{\Delta , \tau }$ . We have

$$ \begin{align*} J_{\Sigma}(\Delta) &= \int\limits_{V} \sum_{\sigma \in \Sigma} (-1)^{\dim \sigma} K_\sigma(x) \, {\textbf 1}_{T^-_{\Delta, \sigma}}(x) dx \\ &= \int\limits_{V} \sum_{\sigma \in \Sigma} (-1)^{\dim \sigma} K_\sigma(x) \left( \sum_{\tau: \tau \preceq \sigma} (-1)^{\dim \tau} {\textbf 1}_{C_{\sigma}^{\tau}}(x) \, \gamma_{\Delta, \tau}(x) \right) dx \\ &= \sum_{\tau \in \Sigma} (-1)^{\dim \tau} \int\limits_{V} \left( \sum_{\sigma: \tau \preceq \sigma} (-1)^{\dim \sigma} K_\sigma(x) \, {\textbf 1}_{C_{\sigma}^{\tau}}(x)\, \gamma_{\Delta, \tau}(x) \right) dx. \end{align*} $$

Now, we use the assumption that $K_\sigma (x)$ is invariant along $\sigma $ and $\gamma _{\Delta , \tau }$ is invariant along $\tau ^\perp $ (by definition of $\gamma _{\Delta , \tau }$ ) to write the above as

$$ \begin{align*}\sum_{\tau} (-1)^{\dim \tau} \left( \, \int\limits_{\tau^\perp} \sum_{\sigma: \tau \preceq \sigma} (-1)^{\dim \sigma} K_\sigma(x_2)\, {\textbf 1}_{C_{\sigma}^{\tau}}(x_2) dx_2 \right) \cdot \left( \, \int\limits_{\operatorname{Span}(\tau)} \gamma_{\Delta, \tau}(x_1) dx_1 \right). \end{align*} $$

Here, $x=x_1 + x_2$ where $x_1 \in \operatorname {Span}(\tau )$ and $x_2 \in \tau ^\perp $ , and $dx_1$ and $dx_2$ are the Lebesgue measures on $\operatorname {Span}(\tau )$ and $\tau ^\perp $ , respectively, so that $dx=dx_1 dx_2$ . By Theorem 2.1 and Remark 2.9, we know that

$$ \begin{align*} \operatorname{vol}(\Gamma_{\Delta, \tau}) = \int\limits_{\operatorname{Span}(\tau)} \gamma_{\Delta, \tau}(x_1) \, dx_1 \end{align*} $$

is a polynomial in the support numbers of $\Gamma _{\Delta , \tau }$ of degree $\dim \tau $ . By definition (see (4.1)), these support numbers either correspond to the $b_i$ in which case they are $0$ , or they correspond to the $w_i$ in which case they are equal to the $a_i$ , the corresponding support numbers of $\Delta $ . It follows that $\operatorname {vol}(\Gamma _{\Delta , \tau })$ is a polynomial in the support numbers of $\Delta $ of degree $\dim \tau $ . Recall that the normal fan of the face of $\Delta $ corresponding to $\tau $ is the fan $\Sigma / \tau $ consisting of all the images of the cones $\sigma \succeq \tau $ in the quotient vector space $V / \operatorname {Span}(\tau ) \cong \tau ^\perp $ . One then observes that $\int \limits _{\tau ^\perp } \sum \limits _{\sigma : \tau \preceq \sigma } (-1)^{\dim \sigma } K_\sigma (x)\, {\textbf 1}_{C_{\sigma }^{\tau }}(x) \, dx_2$ is exactly $J_{\Sigma /\tau }(0)$ . In summary,

$$ \begin{align*} J_{\Sigma}(\Delta) = \sum_{\tau \in \Sigma} (-1)^{\dim \tau} J_{\Sigma/\tau}(0) \operatorname{vol}(\Gamma_{\Delta, \tau}). \end{align*} $$

This shows that $J_{\Sigma }(\Delta )$ is a linear combination of the polynomials $\operatorname {vol}(\Gamma _{\Delta , \tau })$ and hence is a polynomial itself. It remains to show that $J_{\Sigma /\tau }(0)$ is convergent. But this is the content of Lemma 3.7, and the proof is finished.▪

Proof of Theorem 4.2

In the proof of Theorem 4.1, replace any integral $\int \limits _A f(x)dx$ with a sum $\sum \limits _{m \in A \cap M} f(m)$ . In particular, replace $\operatorname {vol}$ with the number of lattice points. For $\tau \in \Sigma $ , let $M_1 = \operatorname {Span}(\tau ) \cap M$ and $M_2 = \tau ^\perp \cap M$ . Note that it is possible that $M_1 + M_2 \neq M$ . Nevertheless, $M_1+M_2$ is a subgroup of finite index in M. Let $M' \subset M$ be a system of coset representatives for $M/ (M_1+M_2)$ . Then every $m \in M$ can be uniquely written as $m'+m_1+m_2$ where $m' \in M'$ and $m_i \in M_i$ . Then, similar to the proof of Theorem 4.1, we write

$$ \begin{align*} S_{\Sigma}(\Delta, M) &= \sum_{m \in M} \sum_{\sigma \in \Sigma} (-1)^{\dim \sigma} K_\sigma(m) \, {\textbf 1}_{T^-_{\Delta, \sigma}}(m)\\ &= \sum_{m \in M} \sum_{\sigma \in \Sigma} (-1)^{\dim \sigma} K_\sigma(m) \left(\sum_{\tau: \tau \preceq \sigma} (-1)^{\dim \tau} {\textbf 1}_{C_{\sigma}^{\tau}}(m) \, \gamma_{\Delta, \tau}(m)\right)\\ &= \sum_{\tau \in \Sigma} (-1)^{\dim \tau} \sum_{m \in M} \left(\sum_{\sigma: \tau \preceq \sigma} (-1)^{\dim \sigma} K_\sigma(m)\, {\textbf 1}_{C_{\sigma}^{\tau}}(m)\, \gamma_{\Delta, \tau}(m)\right) \nonumber \\ &\begin{aligned} =\sum_{\tau} (-1)^{\dim \tau} \sum_{m' \in M'} &\left( \sum_{m_2 \in M_2} \sum_{\sigma: \tau \preceq \sigma} (-1)^{\dim \sigma} K_\sigma(m'+m_2)\, {\textbf 1}_{C_{\sigma}^{\tau}}(m'+m_2) \right) \\ &\cdot \left( \sum_{m_1 \in M_1} \gamma_{\Delta, \tau}(m'+m_1) \right). \end{aligned} \end{align*} $$

One shows that, for fixed $m' \in M'$ , the quantity $\sum \limits _{m_2 \in M_2} \sum \limits _{\sigma : \tau \preceq \sigma } (-1)^{\dim \sigma } K_\sigma (m'+m_2)$ is equal to $S_{\Sigma /\tau }(0)$ with respect to the functions $K_\sigma (m'+x)$ (instead of $K_\sigma (x)$ ). By the discrete version of Lemma 3.7, we know that $S_{\Sigma /\tau }(0)$ is convergent. Let us see that the other term $\sum \limits _{m_1 \in M_1} \gamma _{\Delta , \tau }(m'+m_1)$ depends polynomially on $\Delta $ . Let $\pi : V \to \operatorname {Span}(\tau )$ be the orthogonal projection. Since $\gamma _{\Delta , \tau }$ is invariant in the $\tau ^\perp $ direction, we have $\gamma _{\Delta , \tau }(m'+m_1) = \gamma _{\Delta , \tau }(\pi (m')+m_1)$ . Now, the polynomiality of $\sum \limits _{m_1 \in M} \gamma _{\Delta , \tau }(\pi (m')+m_1)$ follows from Remark 2.9 (see also Theorem 2.3 and Remark 2.4). Thus, $S_{\Sigma }(\Delta , M)$ is a finite sum (over $m' \in M'$ ) of polynomials and hence a polynomial itself. This finishes the proof.▪

5.1 Background on toric varieties

In this section, we review some basic facts about toric varieties. Common references on toric varieties are [Reference Cox, Little and SchenckCLS11, Reference FultonFu93]. Let $T = T_N \cong ({\mathbb C}^*)^n$ be an algebraic torus of dimension n over ${\mathbb C}$ , with character lattice $M \cong {\mathbb Z}^n$ and cocharacter lattice $N \cong {\mathbb Z}^n$ . We denote the corresponding vector spaces $N \otimes _{\mathbb Z} {\mathbb R}$ and $M \otimes _{\mathbb Z} {\mathbb R}$ by $N_{\mathbb R}$ and $M_{\mathbb R}$ , respectively. For $m \in M$ , we denote the corresponding character/irreducible representation by $\chi _m: T \to {\mathbb C}^*$ .

Let $\sigma \subset N_{\mathbb R}$ be a rational strongly convex polyhedral cone. Recall that $\sigma $ is rational if it is generated as a cone by vectors from N. To $\sigma $ , one associates an affine toric variety $U_\sigma $ defined by

$$ \begin{align*} U_\sigma = \operatorname{Spec}({\mathbb C}[\sigma^\vee \cap M]. \end{align*} $$

Here, ${\mathbb C}[\sigma ^\vee \cap M]$ is the semigroup algebra of the semigroup of all lattice points in the dual cone $\sigma ^\vee $ . If $\tau \preceq \sigma $ , then we have natural inclusion $U_\tau \hookrightarrow U_\sigma $ . The variety $U_0$ associated with the origin is just the algebraic torus T itself. The M-grading on the algebra ${\mathbb C}[\sigma ^\vee \cap M]$ induces a T-action on the variety $U_\sigma $ with open orbit $U_0$ .

Recall that a fan $\Sigma $ in $N_{\mathbb R}$ is rational if all the cones in $\Sigma $ are generated by vectors in N. Let $X_{\Sigma }$ be the toric variety corresponding to a complete rational fan $\Sigma $ (see [Reference Cox, Little and SchenckCLS11, Chapter 3] for more details). The (abstract) variety $X_{\Sigma }$ is obtained by gluing all the affine toric varieties $U_\sigma $ , $\sigma \in \Sigma $ , with respect to inclusion maps $U_\tau \hookrightarrow U_\sigma $ and $\tau \preceq \sigma $ .

There is an inclusion-reversing correspondence between the cones in $\Sigma $ and the T-orbits in $X_{\Sigma }$ . For $\sigma \in \Sigma $ , let the corresponding T-orbit be $O_\sigma $ .

For a ray $\rho \in \Sigma (1)$ , we denote the corresponding T-orbit closure $\overline {O}_\rho $ by $D_\rho $ . The $D_\rho $ for $\rho \in \Sigma (1)$ are T-invariant prime divisors on $X_{\Sigma }$ . For each ray $\rho \in \Sigma (1)$ , let $v_\rho \in N$ be the primitive vector along $\rho $ , i.e., shortest lattice vector on $\rho $ . Let $\xi \in \sigma \cap N$ be a cocharacter. One knows that for $x \in U_0$ , $\lim _{t \to 0} \xi (t) \cdot x$ exists and is a point in the orbit $O_\sigma $ .

Let us assume that $X_{\Sigma }$ is a projective variety. This is equivalent to the set $\mathcal {P}(\Sigma )$ , of polytopes with normal fan $\Sigma $ , being nonempty. Let $\Delta \subset M_{\mathbb R}$ be a lattice polytope with normal fan $\Sigma $ . The faces of $\Delta $ are in one-to-one correspondence with cones in $\Sigma $ . For $\sigma \in \Sigma $ , let $Q_\sigma $ be the corresponding face of $\Delta $ . We note that $\dim Q_\sigma = \operatorname {codim} \sigma $ . The polytope $\Delta $ can be represented as