7.1. Integration on polytopes [02X2]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

7.1. Integration on polytopes

In this section, we present a closed formula for the integral over a polytope of a function of one variable composed with a linear form, extending in this direction Brion’s formula for the case of a simplex [Bri88], see Proposition 7.3 and Corollary 7.14 below. In the next section, these formulae will allow us to compute the height of toric varieties with respect to some interesting metrics arising from polytopes.

Let $\Delta\subset\mathbb{R}^{n}$ be a polytope of dimension $n$ and $u\in\mathbb{R}^{n}$ a vector. An aggregate of $\Delta$ in the direction $u$ is defined as the union of the faces of $\Delta$ lying in some affine hyperplane orthogonal to $u$ , provided that the union is non-empty. We write $\Delta(u)$ for the set of aggregates of $\Delta$ in the direction $u$ . Note that, for $V\in\Delta(u)$ and $x$ a point in the affine space spanned by $V$ , the value $\langle u,x\rangle$ is independent of $x$ . We denote this common value by $\langle u,V\rangle$ . For any two aggregates $V_{1},V_{2}\in\Delta(u)$ , we have $V_{1}=V_{2}$ if and only if $\langle u,V_{1}\rangle=\langle u,V_{2}\rangle$ .

In each facet $F$ of $\Delta$ we choose a point $m_{F}$ . Let $L_{F}$ be the linear hyperplane defined by $F$ . Hence, $F-m_{F}$ is a polytope in $L_{F}$ of full dimension $n-1$ . Observe that, for $V\in\Delta(u)$ , the intersection $V\cap F$ is an aggregate of $F$ . We write $\pi_{F}$ for the orthogonal projection of $\mathbb{R}^{n}$ onto $L_{F}$ . We also denote by $u_{F}$ the vector inner normal to $F$ of norm 1.

Definition 7.1.

For each aggregate $V\in\Delta(u)$ , we define the polynomial

C(\Delta,u,V)=\sum_{k=0}^{\dim(V)}\frac{k!}{\dim(V)!}C_{k}(\Delta,u,V)z^{\dim(V)-k}\in\mathbb{R}[z]

recursively. For $k>\dim(V)$ we set $C_{k}(\Delta,u,V)=0$ . For convenience, we set $C(\Delta,u,\emptyset)=0$ for all $\Delta$ and $u$ . If $u=0$ , then $V=\Delta$ and we define $C_{n}(\Delta,0,\Delta)$ as the Lebesgue measure of $\Delta$ and $C_{k}(\Delta,0,\Delta)=0$ , for $k<n$ . If $u\neq 0$ , we set

(7.2)

C_{k}(\Delta,u,V)=-\sum_{F}\frac{\langle u_{F},u\rangle}{\|u\|^{2}}C_{k}(F,\pi_{F}(u),V\cap F),

where the sum is over the facets $F$ of $\Delta$ .

As usual, we write $\mathscr{C}^{n}(\mathbb{R})$ for the space of functions of one real variable which are $n$ -times continuously differentiable. For $f\in\mathscr{C}^{n}(\mathbb{R})$ and $k\geq 0$ , we write $f^{(k)}$ for the $k$ -th derivative of $f$ . Write $\operatorname{vol}_{n}$ for the Lebesgue measure of $\mathbb{R}^{n}$ .

We want to give a formula that, for $f\in\mathscr{C}^{n}(\mathbb{R})$ , computes $\int_{\Delta}f^{(n)}(\langle u,x\rangle)\,\text{\rm d}\operatorname{vol}_{n}$ in terms of the values of $f\circ u$ at the vertices of $\Delta$ . However, when $u$ is orthogonal to some faces of $\Delta$ of positive dimension, such a formula necessarily depends on the values of the derivatives of $f$ .

Proposition 7.3.

Let $\Delta\subset\mathbb{R}^{n}$ be a polytope of dimension $n$ and $u\in\mathbb{R}^{n}$ . Then, for any $f\in\mathscr{C}^{n}(\mathbb{R})$ ,

	$\displaystyle\int_{\Delta}f^{(n)}(\langle u,x\rangle)\,\text{\rm d}\operatorname{vol}_{n}$	$\displaystyle=\sum_{V\in\Delta(u)}\big(C(\Delta,u,V)(z)\cdot f(z+\langle u,V\rangle)\big)^{(\dim(V))}(0)$
(7.4)			$\displaystyle=\sum_{V\in\Delta(u)}\sum_{k\geq 0}C_{k}(\Delta,u,V)f^{(k)}(\langle u,V\rangle).$

The coefficients $C_{k}(\Delta,u,V)$ are uniquely determined by this identity.

Proof.

In view of Definition 7.1 both formulae in the above statement are equivalent and so it is enough to prove the second one. In case $u=0$ , we have $\Delta(u)=\{\Delta\}$ and formula (7.4) holds because

\int_{\Delta}f^{(n)}(\langle 0,x\rangle)\,\text{\rm d}\operatorname{vol}_{n}=\operatorname{vol}(\Delta)f^{(n)}(0)=\sum_{k\geq 0}C_{k}(\Delta,0,\Delta)f^{(k)}(0),

We prove (7.4) by induction on the dimension $n$ . In case $n=0$ , we have $u=0$ and so the verification reduces to the above one. Hence, we assume $n\geq 1$ and $u\neq 0$ . For short, we write $\,\text{\rm d}x=\,\text{\rm d}x_{1}\wedge\dots\wedge\,\text{\rm d}x_{n}$ . Choose any vector $v\in\mathbb{R}^{n}$ of norm $1$ and such that $\langle u,v\rangle\not=0$ . Performing an orientation-preserving orthonormal change of variables, we may assume $v=(1,0,\dots,0)$ . We have

f^{(n)}(\langle u,x\rangle)\,\text{\rm d}x=\frac{1}{\langle u,v\rangle}\,\text{\rm d}\left(f^{(n-1)}(\langle u,x\rangle)\,\text{\rm d}x_{2}\wedge\dots\wedge\,\text{\rm d}x_{n}\right).

With Stokes’ theorem, we obtain

(7.5)		$\displaystyle\int_{\Delta}f^{(n)}(\langle u,x\rangle)\,\text{\rm d}\operatorname{vol}_{n}$	$\displaystyle=\int_{\Delta}f^{(n)}(\langle u,x\rangle)\,\text{\rm d}x$
		$\displaystyle=\frac{1}{\langle u,v\rangle}\sum_{F}\int_{F}f^{(n-1)}(\langle u,x\rangle)\,\text{\rm d}x_{2}\wedge\dots\wedge\,\text{\rm d}x_{n}.$

where the sum is over the facets $F$ of $\Delta$ , and we equip each facet with the induced orientation.

For each facet $F$ of $\Delta$ , we let $\iota_{u_{F}}(\,\text{\rm d}x)$ be the differential form of order $n-1$ obtained by contracting $\,\text{\rm d}x$ with the vector $u_{F}$ . The form $\,\text{\rm d}x_{2}\wedge\dots\wedge\,\text{\rm d}x_{n}$ is invariant under translations and its restriction to the linear hyperplane $L_{F}$ coincides with $\langle u_{F},v\rangle\iota_{u_{F}}(\,\text{\rm d}x)$ . Therefore,

\int_{F}f^{(n-1)}(\langle u,x\rangle)\,\text{\rm d}x_{2}\wedge\dots\wedge\,\text{\rm d}x_{n}=\langle u_{F},v\rangle\int_{F-m_{F}}f^{(n-1)}(\langle u,x+m_{F}\rangle)\iota_{u_{F}}(\,\text{\rm d}x).

Let $\operatorname{vol}_{n-1}$ denote the Lebesgue measure on $L_{F}$ . We can verify that $\operatorname{vol}_{n-1}$ coincides with the measure induced by integration of $-\iota_{u_{F}}(\,\text{\rm d}x)$ along $L_{F}$ . Let $g\colon\mathbb{R}\to\mathbb{R}$ be the function defined as $g(z)=f(z+\langle u,m_{F}\rangle)$ . Then $f^{(n-1)}(\langle u,x+m_{F}\rangle)=g^{(n-1)}(\langle\pi_{F}(u),x\rangle)$ for all $x\in L_{F}$ . Hence,

\int_{F-m_{F}}f^{(n-1)}(\langle u,x+m_{F}\rangle)\iota_{u_{F}}(\,\text{\rm d}x)=-\int_{F-m_{F}}g^{(n-1)}(\langle\pi_{F}(u),x\rangle)\,\text{\rm d}\operatorname{vol}_{n-1}.

Applying the inductive hypothesis to $F$ and the function $g$ we obtain

	$\displaystyle\int_{F}g^{(n-1)}(\langle\pi_{F}(u),x\rangle)\,\text{\rm d}\operatorname{vol}_{n-1}$	$\displaystyle=\sum_{V^{\prime}\in F(\pi_{F}(u))}\sum_{k\geq 0}C_{k}(F,\pi_{F}(u),V^{\prime})g^{(k)}(\langle\pi_{F}(u),V^{\prime}\rangle)$
		$\displaystyle=\sum_{V^{\prime}\in F(\pi_{F}(u))}\sum_{k\geq 0}C_{k}(F,\pi_{F}(u),V^{\prime})f^{(k)}(\langle u,V^{\prime}\rangle).$

Each aggregate $V^{\prime}\in F(\pi_{F}(u))$ is contained in a unique $V\in\Delta(u)$ and it coincides with $V\cap F$ . Therefore, we can transform the right-hand side of the last equality in

\sum_{V\in\Delta(u)}\sum_{k\geq 0}C_{k}(F,\pi_{F}(u),V\cap F)f^{(k)}(\langle u,V\rangle),

where, for simplicity, we have set $C_{k}(F,\pi_{F}(u),V\cap F)=0$ whenever $V\cap F=\emptyset$ . Plugging the resulting expression into (7.5) and exchanging the summations on $V$ and $F$ , we obtain that $\int_{\Delta}f^{(n)}(\langle x,u\rangle)\,\text{\rm d}\operatorname{vol}_{n}$ is equal to

(7.6)

\sum_{V\in\Delta(u)}\sum_{k\geq 0}\bigg(-\sum_{F}\frac{\langle u_{F},v\rangle}{\langle u,v\rangle}C_{k}(F,\pi_{F}(u),V\cap F)f^{(k)}(\langle u,V\rangle)\bigg).

Specialising this identity to $v=u$ , we readily derive formula (7.4) from Definition 7.1 of the coefficients $C_{k}(\Delta,u,V)$ .

For the last statement, observe that the values $f^{(k)}(\langle u,V\rangle)$ can be arbitrarily chosen. Hence, the coefficients $C_{k}(\Delta,u,V)$ are uniquely determined from the linear system obtained from the identity (7.4) for enough functions $f$ . ∎

Corollary 7.7.

Let $\Delta\subset\mathbb{R}^{n}$ be a polytope of dimension $n$ and $u\in\mathbb{R}^{n}$ . Then,

\operatorname{vol}_{n}(\Delta)=\sum_{V\in\Delta(u)}\sum_{k=0}^{\dim(V)}C_{k}(\Delta,u,V)\frac{\langle u,V\rangle^{n-k}}{(n-k)!}.

Proof.

This follows from formula (7.4) applied to the function $f(z)=z^{n}/n!$ . ∎

Proposition 7.8.

Let $\Delta\subset\mathbb{R}^{n}$ be a polytope of dimension $n$ and $u\in\mathbb{R}^{n}$ . Let $V\in\Delta(u)$ and $k\geq 0$ .

(1)

The coefficient $C_{k}(\Delta,u,V)$ is homogeneous of weight $k-n$ , in the sense that, for $\lambda\in\mathbb{R}^{\times}$ ,

$C_{k}(\Delta,\lambda u,V)=\lambda^{k-n}C_{k}(\Delta,u,V).$
(2)

The coefficients $C_{k}(\Delta,u,V)$ satisfy the vector relation

(7.9) $C_{k}(\Delta,u,V)\cdot u=-\sum_{F}C_{k}(F,\pi_{F}(u),V\cap F)\cdot u_{F},$

where the sum is over the facets $F$ of $\Delta$ .
(3)

Let $\Delta_{1},\Delta_{2}\subset\mathbb{R}^{n}$ be two polytopes of dimension $n$ intersecting along a common facet and such that $\Delta=\Delta_{1}\cup\Delta_{2}$ . Then $V\cap\Delta_{i}=\emptyset$ or $V\cap\Delta_{i}\in\Delta_{i}(u)$ and

$C_{k}(\Delta,u,V)=C_{k}(\Delta_{1},u,V\cap\Delta_{1})+C_{k}(\Delta_{2},u,V\cap\Delta_{2}).$

Proof.

Statement (1) follows easily from the definition of $C_{k}(\Delta,u,V)$ . For statement (2), we use that, from (7.6), the integral formula in Proposition 7.3 also holds for the choice of coefficients

-\sum_{F}\frac{\langle u_{F},v\rangle}{\langle u,v\rangle}C_{k}(F,\pi_{F}(u),V\cap F)

for any vector $v$ of norm 1 such that $\langle u,v\rangle\neq 0$ . But the coefficients satisfying that formula are unique. Hence, this choice necessarily coincides with $C_{k}(\Delta,u,V)$ for all such $v$ . Hence,

\langle u,v\rangle C_{k}(\Delta,u,V)=-\sum_{F}\langle u_{F},v\rangle C_{k}(F,\pi_{F}(u),V\cap F)

and formula (7.9) follows. Statement (3) follows from Formula (7.4) applied to $\Delta$ , $\Delta_{1}$ and $\Delta_{2}$ together with the additivity of the integral and the fact that the coefficients $C_{k}(\Delta,u,V)$ are uniquely determined. ∎

Example 7.10.

In case $\Delta$ is a simplex, its aggregates in a given direction $u\in\mathbb{R}^{n}$ are some of its faces and the corresponding coefficients can be made explicit. Indeed, they satisfy the linear system

\sum_{V\in\Delta(u)}\sum_{k=0}^{\min\{i,\dim(V)\}}C_{k}(\Delta,u,V)\frac{\langle u,V\rangle^{i-k}}{(i-k)!}=\begin{cases}0&\mbox{for }i=0,\dots,n-1,\\ {\rm Vol}_{n}(\Delta)&\mbox{for }i=n.\end{cases}

This system has as many unknowns as equations and might be solved using Cramer’s rule. These coefficients admit the closed formula below, which the reader might check using the recurrence relation (7.9):

(7.11)

C_{k}(\Delta,u,V)=(-1)^{\dim(V)-k}\frac{n!}{k!}{\operatorname{vol}}_{n}(\Delta)\sum_{|\beta|=\dim(V)-k}\prod_{\nu\notin V}\langle V-\nu,u\rangle^{-\beta_{\nu}-1},

where the products are over the vertices $\nu$ of $\Delta$ not lying in $V$ and the sum is over the tuples $\beta$ of non negative integers of length $\dim(V)-k$ , indexed by those same vertices of $\Delta$ that are not in $V$ , that is, $\beta\in\mathbb{N}^{n-\dim(V)}$ and $|\beta|=\dim(V)-k$ . In case $V=\nu_{0}$ is a vertex of $\Delta$ , the above formula reduces to

(7.12)

C_{0}(\Delta,u,\nu_{0})=n!{\operatorname{vol}}_{n}(\Delta)\prod_{\nu\neq\nu_{0}}\langle\nu_{0}-\nu,u\rangle^{-1}.

Suppose that the simplex is presented as the intersection of $n+1$ halfspaces as $\Delta=\bigcap_{i=0}^{n}\{x\in\mathbb{R}^{n}|\,\langle u_{i},x\rangle-\lambda_{i}\geq 0\}$ for some $u_{i}\in\mathbb{R}^{n}$ and $\lambda_{i}\in\mathbb{R}$ . Up to a reordering, we can assume that $u_{0}$ is normal to the unique face of $\Delta$ not containing $\nu_{0}$ and that $\det(u_{1},\dots,u_{n})>0$ . Then the above coefficient can be alternatively written as

(7.13)

C_{0}(\Delta,u,\nu_{0})=\frac{\det(u_{1},\dots,u_{n})^{n-1}}{\prod_{i=1}^{n}\det(u_{1},\dots,u_{i-1},u,u_{i+1},\dots,u_{n})}.

We obtain the following extension of Brion’s ‘‘short formula’’ for the case of a simplex [Bri88, Théorème 3.2], see also [BBDL+11].

Corollary 7.14.

Let $\Delta\subset\mathbb{R}^{n}$ be a simplex of dimension $n$ that is the convex hull of points $\nu_{i}$ , $i=0,\dots,n$ , and let $u\in\mathbb{R}^{n}$ such that $\langle u,\nu_{i}\rangle\neq\langle u,\nu_{j}\rangle$ for $i\neq j$ . Then, for any $f\in\mathscr{C}^{n}(\mathbb{R})$ ,

\displaystyle\int_{\Delta}f^{(n)}(\langle u,x\rangle)\,\text{\rm d}\operatorname{vol}_{n}=n!{\operatorname{vol}}_{n}(\Delta)\sum_{i=0}^{n}\frac{f(\langle u,\nu_{i}\rangle)}{\prod_{j\neq i}\langle\nu_{i}-\nu_{j},u\rangle}.

Proof.

This follows from Proposition 7.3 and equation (7.12). ∎

In the next section, we will have to compute integrals over a polytope of functions of the form $\ell(x)\log(\ell(x))$ where $\ell$ is an affine function. The following result gives the value of such integral for the case of a simplex.

Proposition 7.15.

Let $\Delta\subset\mathbb{R}^{n}$ be a simplex of dimension $n$ and let $\ell\colon\mathbb{R}^{n}\to\mathbb{R}$ be an affine function which is non-negative on $\Delta$ . Write $\ell(x)=\langle u,x\rangle-\lambda$ for some vector $u$ and constant $\lambda$ . Then $\displaystyle\frac{1}{\operatorname{vol}_{n}(\Delta)}\int_{\Delta}\ell(x)\log(\ell(x))\,\text{\rm d}\operatorname{vol}_{n}$ equals

(7.16)

\sum_{V\in\Delta(u)}\sum_{\beta^{\prime}}\binom{n}{n-|\beta^{\prime}|}\frac{\ell(V)\left(\log(\ell(V))-\sum_{j=2}^{|\beta^{\prime}|+1}\frac{1}{j}\right)}{(|\beta^{\prime}|+1)\prod_{\nu\notin V}\left(-\big(\frac{\ell(\nu)}{\ell(V)}-1\big)^{\beta^{\prime}_{\nu}}\right)},

where the second sum is over $\beta^{\prime}\in(\mathbb{N}^{\times})^{n-\dim(V)}$ with $|\beta^{\prime}|\leq n$ and the product is over the $n-\dim(V)$ vertices $\nu$ of $\Delta$ not in $V$ . In case $\ell(x)$ is the defining equation of a hyperplane containing a facet $F$ of $\Delta$ ,

(7.17)

\frac{1}{\operatorname{vol}_{n}(\Delta)}\int_{\Delta}\ell(x)\log(\ell(x))\,\text{\rm d}x=\frac{\ell(\nu_{F})}{n+1}\bigg(\log(\ell(\nu_{F}))-\sum_{j=2}^{n+1}\frac{1}{j}\bigg),

where $\nu_{F}$ denotes the unique vertex of $\Delta$ not contained in $F$ .

Proof.

This follows from formulae (7.4) and (7.11) with the function $f^{(n)}(z)=(z-\lambda)\log(z-\lambda)$ , a $(n-k)$ -th primitive of which is

f^{(k)}(z)=\frac{(z-\lambda)^{n-k+1}}{(n-k+1)!}\left(\log(z-\lambda)-\sum_{j=2}^{n-k+1}\frac{1}{j}\right).

∎

We end this section with a lemma specific to integration on the standard simplex.

Lemma 7.18.

Let $\Delta^{r}$ be the standard simplex of $\mathbb{R}^{r}$ and $\beta=(\beta_{0},\dots,\beta_{r-1})\in\mathbb{N}^{r}$ . Let $f\in\mathscr{C}^{|\beta|+r}(\mathbb{R})$ where $|\beta|=\beta_{0}+\dots+\beta_{r-1}$ . For $(w_{1},\dots,w_{r})\in\Delta^{r}$ write $w_{0}=1-w_{1}-\dots-w_{r}$ . Then

\int_{\Delta^{r}}\bigg(\prod_{i=0}^{r-1}\frac{w_{i}^{\beta_{i}}}{\beta_{i}!}\bigg)f^{(|\beta|+r)}(w_{r})\,\text{\rm d}w_{1}\wedge\cdots\wedge\,\text{\rm d}w_{r}=f(1)-\sum_{j=0}^{|\beta|+r-1}\frac{f^{(j)}(0)}{j!}.

Proof.

We proceed by induction on $r$ . Let $r=1$ . Applying $\beta_{0}+1$ successive integrations by parts, the integral computes as

\sum_{j=0}^{\beta_{0}}\left[\frac{(1-w_{1})^{j}}{j!}f^{(j)}(w_{1})\right]_{0}^{1}=f(1)-\sum_{j=0}^{\beta_{0}}\frac{f^{(j)}(0)}{j!},

as stated. Let $r\geq 2$ . Applying the case $r-1$ to the function $f(z)=\frac{z^{|\beta|+r-1}}{(|\beta|+r-1)!}$ ,

\frac{1}{\beta_{0}!\dots\beta_{r-1}!}\int_{\Delta_{r-1}}w_{0}^{\beta_{0}}w_{1}^{\beta_{1}}\dots w_{r-1}^{\beta_{r-1}}\,\text{\rm d}w_{1}\wedge\dots\wedge\,\text{\rm d}w_{r-1}=\frac{1}{(|\beta|+r-1)!}

and, after rescaling,

\frac{1}{\beta_{0}!\dots\beta_{r-1}!}\int_{(1-w_{r})\Delta_{r-1}}w_{0}^{\beta_{0}}w_{1}^{\beta_{1}}\dots w_{r-1}^{\beta_{r-1}}\,\text{\rm d}w_{1}\wedge\dots\wedge\,\text{\rm d}w_{r-1}=\frac{(1-w_{r})^{|\beta|+r-1}}{(|\beta|+r-1)!}.

Therefore, the left-hand side of the equality to be proved reduces to

\frac{1}{(|\beta|+r-1)!}\int_{0}^{1}(1-w_{r})^{|\beta|+r-1}f^{(|\beta|+r)}(w_{r})\,\text{\rm d}w_{r}.

Applying the case $r=1$ and index $|\beta|+r-1\in\mathbb{N}$ , we find that this integral equals $f(1)-\sum_{j=0}^{|\beta|+r-1}f^{(j)}(0)/j!$ , which concludes the proof. ∎

Corollary 7.19.

Let $\alpha\in\mathbb{N}^{r+1}$ . For $(w_{1},\dots,w_{r})\in\Delta^{r}$ , write $w_{0}=1-w_{1}-\dots-w_{r}$ . Then

\int_{\Delta^{r}}w_{0}^{\alpha_{0}}w_{1}^{\alpha_{1}}\dots w_{r}^{\alpha_{r}}\,\text{\rm d}w_{1}\wedge\dots\wedge\,\text{\rm d}w_{r}=\frac{\alpha_{0}!\dots\alpha_{r}!}{(|\alpha|+r)!}

and, for $i=0,\dots,r$ ,

\int_{\Delta^{r}}w_{0}^{\alpha_{0}}w_{1}^{\alpha_{1}}\dots w_{r}^{\alpha_{r}}\log(w_{i})\,\text{\rm d}w_{1}\wedge\dots\wedge\,\text{\rm d}w_{r}=-\frac{\alpha_{0}!\dots\alpha_{r}!}{(|\alpha|+r)!}\sum_{j=\alpha_{i}+1}^{|\alpha|+r}\frac{1}{j}.

Proof.

The first integral follows from Lemma 7.18 applied to $\beta=(\alpha_{0},\dots,\alpha_{r-1})$ and $f(z)=\frac{z^{|\alpha|+r}}{(|\alpha|+r)!}$ . The second one follows similarly, applying Lemma 7.18 to the function $f(z)=\frac{z^{|\alpha|+r}}{(|\alpha|+r)!}\left(\log(z)-\sum_{j=\alpha_{i}+1}^{|\alpha|+r}\frac{1}{j}\right)$ , after some possible permutation (for $i=1,\dots,r-1$ ) or linear change of variables (for $i=0$ ). ∎