7. Metrics from polytopes [02X1]

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7. Metrics from polytopes

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 $i=0$ ). ∎

7.2. Metrics, heights and entropy

In this section we will consider some metrics arising from polytopes. We will use the notation of §4 and §5. In particular, we consider a split torus over the field of rational numbers $\mathbb{T}\simeq\mathbb{G}_{m,\mathbb{Q}}^{n}$ and we denote by $N,M,N_{\mathbb{R}},M_{\mathbb{R}}$ the lattices and dual spaces corresponding to $\mathbb{T}$ .

Let $\Delta\subset M_{\mathbb{R}}$ be a lattice polytope of dimension $n$ . Let $\ell_{i}$ , $i=1,\dots,r$ , be affine functions on $M_{\mathbb{R}}$ defined as $\ell_{i}(x)=\langle u_{i},x\rangle-\lambda_{i}$ for some $u_{i}\in N_{\mathbb{R}}$ and $\lambda_{i}\in\mathbb{R}$ such that $\ell_{i}\geq 0$ on $\Delta$ and let also $c_{i}>0$ . Write $\ell=(\ell_{1},\dots,\ell_{r})$ and $c=(c_{1},\dots,c_{r})$ . We consider the function $\vartheta_{\Delta,\ell,c}\colon\Delta\to\mathbb{R}$ defined, for $x\in\Delta$ , by

(7.20)

\vartheta_{\Delta,\ell,c}(x)=-\sum_{i=1}^{r}c_{i}\ell_{i}(x)\log(\ell_{i}(x)).

When $\Delta,\ell,c$ are clear from the context, we write for short $\vartheta=\vartheta_{\Delta,\ell,c}$ .

Lemma 7.21.

Let notation be as above.

(1)

The function $\vartheta_{\Delta,\ell,c}$ is concave.
(2)

If the family $\{u_{i}\}_{i}$ generates $N_{\mathbb{R}}$ , then $\vartheta_{\Delta,\ell,c}$ is strictly concave.
(3)

If $\Delta=\bigcap_{i}\{x\in M_{\mathbb{R}}|\ell_{i}(x)\geq 0\}$ , then the restriction of $\vartheta_{\Delta,\ell,c}$ to $\Delta^{\circ}$ is of Legendre type (Definition 3.51).

Proof.

Let $1\leq i\leq r$ and consider the affine map $\ell_{i}\colon\Delta\to\mathbb{R}_{\geq 0}$ . We have that $-z\log(z)$ is a strictly concave function on $\mathbb{R}_{\geq 0}$ and $-\ell_{i}\log(\ell_{i})=\ell_{i}^{*}(-z\log(z))$ . Hence, each function $-c_{i}\ell_{i}(x)\log(\ell_{i}(x))$ is concave and so is $\vartheta$ , as stated in (1)

For statement (2), let $x_{1},x_{2}$ be two different points of $\Delta$ . The assumption that $\{u_{i}\}_{i}$ generates $N_{\mathbb{R}}$ implies that $\ell_{i_{0}}(x_{1})\neq\ell_{i_{0}}(x_{2})$ for some $i_{0}$ . Hence, the affine map $\ell_{i_{0}}$ gives an injection of the segment ${\overline{x_{1}x_{2}}}$ into $\mathbb{R}_{\geq 0}$ . We deduce that $-c_{i_{0}}\ell_{i_{0}}\log(\ell_{i_{0}})$ is strictly concave on ${\overline{x_{1}x_{2}}}$ and so is $\vartheta$ . Varying $x_{1},x_{2}$ , we deduce that $\vartheta$ is strictly concave on $\Delta$ .

For statement (3), it is clear that $\vartheta|_{\Delta^{\circ}}$ is differentiable. Moreover, the assumption that $\Delta$ is the intersection of the halfspaces defined by the $\ell_{i}$ ’s implies that the $u_{i}$ ’s generate $N_{\mathbb{R}}$ and so $\vartheta$ is strictly concave. The gradient of $\vartheta$ is given, for $x\in\Delta^{\circ}$ , by

(7.22)

\nabla\vartheta(x)=-\sum_{i=1}^{r}c_{i}u_{i}(\log(\ell_{i}(x)+1).

Let $\|\cdot\|$ be a fixed norm on $M_{\mathbb{R}}$ and $(x_{j})_{j\geq 0}$ a sequence in $\Delta^{\circ}$ converging to a point in the border. Then there exists some $i_{1}$ such $\ell_{i_{1}}(x_{j})\stackrel{{\scriptstyle j}}{{\to}}0$ . Thus, $\|\nabla\vartheta(x)\|\stackrel{{\scriptstyle j}}{{\to}}\infty$ and the statement follows. ∎

Definition 7.23.

Let $\Sigma_{\Delta}$ and $\Psi_{\Delta}$ be the fan and the support function on $N_{\mathbb{R}}$ induced by $\Delta$ . Let $(X_{\Sigma_{\Delta}},D_{\Psi_{\Delta}})$ be the associated polarized toric variety over $\mathbb{Q}$ and write $L={\mathcal{O}}(D_{\Psi_{\Delta}})$ . By Lemma 7.21(1), $\vartheta$ is a concave function on $\Delta$ . By Theorem 5.73, it corresponds to some approachable toric metric on $L(\mathbb{C})$ . We denote this metric by $\|\cdot\|_{\Delta,\ell,c}$ . We write ${\overline{L}}$ for the line bundle $L$ equipped with the metric $\|\cdot\|_{\Delta,\ell,c}$ at the Archimedean place of $\mathbb{Q}$ and with the canonical metric at the non-Archimedean places. This is an example of an adelic toric metric.

Example 7.24.

Following the notation in Example 3.53, consider the standard simplex $\Delta^{n}$ and the concave function $\vartheta=\frac{1}{2}\varepsilon_{n}$ on $\Delta^{n}$ . From examples 3.53 and 5.18(1), we deduce that the corresponding metric is the Fubini-Study metric of ${\mathcal{O}}(1)^{{\text{\rm an}}}$ .

In case $\Delta$ is the intersection of the halfspaces defined by the $\ell_{i}$ ’s, Lemma 7.21(3) shows that $\vartheta|_{\Delta^{\circ}}$ of Legendre type (Definition 3.51). By Theorem 3.52 and equation (7.22), the gradient of $\vartheta$ gives a homeomorphism between $\Delta^{\circ}$ and $N_{\mathbb{R}}$ and, for $x\in\Delta^{\circ}$ ,

(7.25)

\vartheta^{\vee}(\nabla\vartheta(x))=-\sum_{i=1}^{r}c_{i}\lambda_{i}\log(\ell_{i}(x))+c_{i}\langle u_{i},x\rangle.

This gives an explicit expression of the function $\psi_{\|\cdot\|_{\Delta,\ell,c}}=\vartheta^{\vee}$ , and a fortiori of the metric ${\|\cdot\|_{\Delta,\ell,c}}$ , in the coordinates of the polytope. Up to our knowledge, there is no simple expression for $\psi$ in linear coordinates of $N_{\mathbb{R}}$ , except for special cases like Fubini-Study.

Remark 7.26.

This kind of metrics are interesting when studying the Kähler geometry of toric varieties. Given a Delzant polytope $\Delta\subset M_{\mathbb{R}}$ , Guillemin has constructed a “canonical” Kähler structure on the associated symplectic toric variety [Gui95]. The corresponding symplectic potential is the function $-\vartheta_{\Delta,\ell,c}$ , for the case when $r$ is the number of facets of $\Delta$ , $c_{i}=1/2$ for all $i$ , and $u_{i}$ is a primitive vector in $N$ and $\lambda_{i}$ is an integer such that $\Delta=\{x\in M_{\mathbb{R}}|\langle u_{i},x\rangle\geq\lambda_{i},i=1,\dots,r\}$ , see [Gui95, Appendix 2, (3.9)].

In this case, the metric $\|\cdot\|_{\Delta,\ell,c}$ on the line bundle ${\mathcal{O}}(D_{\Psi})^{{\text{\rm an}}}$ is smooth and positive and, as explained in Remark 5.74, its Chern form gives this canonical Kähler form.

We obtain the following formula for the height of $X_{\Sigma_{\Delta}}$ with respect to the adelic metrized line bundle $\overline{L}$ , in terms of the coefficients $C_{k}(\Delta,u_{i},V)$ .

Proposition 7.27.

Let notation be as in Definition 7.23. Then $\operatorname{h}_{\overline{L}}(X_{\Sigma_{\Delta}})$ equals

{(n+1)!}\sum_{i=1}^{r}c_{i}\sum_{V\in\Delta(u_{i})}\sum_{k=0}^{\dim(V)}C_{k}(\Delta,u_{i},V)\frac{\ell_{i}(V)^{n-k+1}}{(n-k+1)!}\left(\sum_{j=2}^{n-k+1}\frac{1}{j}-\log(\ell_{i}(V))\right).

Suppose furthermore that $\Delta\subset\mathbb{R}^{n}$ is a simplex, $r=n+1$ and that $\ell_{i}$ , $i=1,\dots,n+1$ , are affine functions such that $\Delta=\bigcap_{i}\{x\in M_{\mathbb{R}}|\ell_{i}(x)\geq 0\}$ . Then

(7.28)

\operatorname{h}_{\overline{L}}(X_{\Sigma_{\Delta}})=n!\operatorname{vol}_{M}(\Delta)\sum_{i=1}^{n+1}c_{i}\ell_{i}(\nu_{i})\bigg(\sum_{j=2}^{n+1}\frac{1}{j}-\log(\ell_{i}(\nu_{i}))\bigg).

where $\nu_{i}$ is the unique vertex of $\Delta$ not contained in the facet defined by $\ell_{i}$ .

Proof.

The first statement follows readily from Theorem 6.37 and Proposition 7.3 applied to the functions $f_{i}(z)=\left(\log(z-\lambda_{i})-\sum_{j=2}^{n+1}\frac{1}{j}\right)(z-\lambda_{i})^{n+1}/(n+1)!$ . The second statement follows similarly from Proposition 7.15. ∎

Example 7.29.

Let ${\mathcal{O}}(1)$ be the universal line bundle of $\mathbb{P}^{n}$ . The Fubini-Study metric of ${\mathcal{O}}(1)^{{\text{\rm an}}}$ corresponds to the case of the standard simplex, $\ell_{i}(x)=x_{i}$ , $i=1,\dots,n$ and $\ell_{n+1}(x)=1-\sum_{i=1}^{n}x_{i}$ and the choice $c_{i}=1/2$ for all $i$ . Hence we recover from (7.28) the well known expression for the height of $\mathbb{P}^{n}$ with respect to the Fubini-Study metric in [BGS94, Lemma 3.3.1]:

\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(\mathbb{P}^{n})=\frac{n+1}{2}\sum_{j=2}^{n+1}\frac{1}{j}.

Example 7.30.

In dimension $1$ , a polytope is an interval of the form $\Delta=[m_{0},m_{1}]$ for some $m_{i}\in\mathbb{Z}$ . The corresponding roof function in (7.20) writes down, for $x\in[m_{0},m_{1}]$ , as

(7.31)

\vartheta(x)=-\sum_{i=1}^{r}c_{i}\ell_{i}(x)\log(\ell_{i}(x))

for affine function $\ell_{i}=u_{i}x-\lambda_{i}$ which take non negative values on the $\Delta$ and $c_{i}>0$

The polarized toric variety corresponding to $\Delta$ is $\mathbb{P}^{1}$ together with the ample divisor $m_{1}[(0:1)]-m_{0}[(1:0)]$ . Write $L={\mathcal{O}}_{\mathbb{P}^{1}}(x_{1}-x_{0})$ for the associate line bundle and ${\overline{L}}$ for the adelic metrized line bundle corresponding to the function $\vartheta$ . The Legendre-Fenchel dual to $-c_{i}\ell_{i}(x)\log(\ell_{i}(x))$ is the function $f_{i}\colon\mathbb{R}\rightarrow\mathbb{R}$ defined, for $v\in\mathbb{R}$ , by

f_{i}(v)=\frac{\lambda_{i}}{u_{i}}v-c_{i}{\operatorname{e}}^{-1-\frac{v}{{c_{i}u_{i}}}}.

Therefore, the function $\psi=\vartheta^{\vee}$ is the sup-convolution of these function, namely $\psi=f_{1}\boxplus\dots\boxplus f_{m}$ For the height, a simple computation shows that

{\operatorname{h}_{\overline{L}}(\mathbb{P}^{1})}=\int_{m_{0}}^{m_{1}}\vartheta\,\text{\rm d}x=\sum_{i=1}^{r}\frac{c_{i}}{4u_{i}}\Big[\ell_{i}(x)^{2}\left(1-2\log(\ell_{i}(x))\right)\Big]^{m_{1}}_{m_{0}}

In some cases, the height of a toric variety with respect to the metrics constructed above has an interpretation in terms of the average entropy of some natural random processes. Let $\Gamma$ be an arbitrary polytope containing $\Delta$ . For a point $x\in\operatorname{ri}(\Delta)$ , we consider the partition $\Pi_{x}$ of $\Gamma$ which consists of the cones $\eta_{x,F}$ of vertex $x$ and base the relative interior of each proper face $F$ of $\Gamma$ .

We consider $\Gamma$ as a probability space endowed with the uniform probability distribution and $\beta_{x}$ the random variable which, for a point $y\in\Gamma$ , returns the base $F$ of the unique cone $\eta_{x,F}$ it belongs to. Clearly, the probability that a given face $F$ is returned is the ratio of the volume of the cone based on $F$ to the volume of $\Gamma$ . We have ${\operatorname{vol}}_{n}(\eta_{x,F})={n}^{-1}{\operatorname{dist}}(x,F){\operatorname{vol}}_{n-1}(F)$ where, as before, $\operatorname{vol}_{n}$ and $\operatorname{vol}_{n-1}$ denote the Lebesgue measure on $\mathbb{R}^{n}$ and on $L_{F}$ , respectively. Hence,

(7.32)

P(\beta_{x}=F)=\begin{cases}\displaystyle\frac{{\operatorname{dist}}(x,F){\operatorname{vol}}_{n-1}(F)}{n{\operatorname{vol}}_{n}(\Gamma)}&\text{ if }\dim(F)=n-1,\\ 0&\text{ if }\dim(F)\leq n-2.\end{cases}

The entropy of the random variable $\beta_{x}$ is

{\mathcal{E}}(x)=-\sum_{F}P(\beta_{x}=F)\log(P(\beta_{x}=F)),

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

For each facet $F$ of $\Gamma$ we let $u_{F}\in\mathbb{R}^{n}$ be the inner normal vector to $F$ of Euclidean norm $(n-1)!\operatorname{vol}_{n-1}(F)$ and $\lambda_{F}=\Psi_{\Gamma}(u_{F})\in\mathbb{R}$ and consider the affine form $\ell_{F}$ defined as $\ell_{F}(x)=\langle u_{F},x\rangle-\lambda_{F}$ . Hence, $\Gamma=\{x\in M_{\mathbb{R}}|\ell_{F}(x)\geq 0\}$ . Let also $c_{F}=c$ for some constant $c>0$ . By the Minkowski condition, $\sum_{F}u_{F}=0$ . Hence $\sum_{F}\ell_{F}=-\sum_{F}\lambda_{F}$ .

Remark 7.33.

Suppose that $\Gamma$ is a lattice polytope and let $F$ be a facet of $\Gamma$ . Recall that $M(F)$ is the lattice $L_{F}\cap M$ and let $M(F)^{\prime}$ be the sublattice of $M(F)$ generated by the differences of the lattice points in $F$ . Then the vector $u_{F}$ can be alternatively defined as $[M(F):M(F)^{\prime}]$ times the primitive inner normal vector to the facet $F$ .

The concave function $\vartheta=-\sum_{F}{c\,\ell_{F}(x)}\log({\ell_{F}(x)})$ belongs to the class of functions considered in Definition 7.23. Thus, we obtain a line bundle with an adelic toric metric ${\overline{L}}$ on $X_{\Delta}$ . For short, we write $X=X_{\Delta}$ . The following result shows that the average entropy of the random variable $\beta_{x}$ with respect to the uniform distribution on $\Delta$ can be expressed in terms of the height of the toric variety $X$ with respect to ${\overline{L}}$ .

Proposition 7.34.

With the above notation,

\frac{1}{\operatorname{vol}_{n}(\Delta)}\int_{\Delta}{\mathcal{E}}(x)\,\text{\rm d}\operatorname{vol}_{n}=\frac{1}{n!\operatorname{vol}_{n}(\Gamma)}\bigg(\frac{\operatorname{h}_{{\overline{L}}}(X)}{c(n+1)\deg_{L}(X)}-{\log(n!\operatorname{vol}_{n}(\Gamma))}\Big(\sum_{F}\lambda_{F}\Big)\bigg)

where the sum is over the facets $F$ of $\Gamma$ . In particular, if $\Gamma=\Delta$ ,

\frac{1}{\operatorname{vol}_{n}(\Delta)}\int_{\Delta}{\mathcal{E}}(x)\,\text{\rm d}\operatorname{vol}_{n}=\frac{\operatorname{h}_{{\overline{L}}}(X)}{c(n+1)\deg_{L}(X)^{2}}-\frac{\log(\deg_{L}(X))}{\deg_{L}(X)}\Big(\sum_{F}\lambda_{F}\Big).

Proof.

For $x\in\operatorname{ri}(\Delta)$ and $F$ a facet of $\Gamma$ , we deduce from equation (7.32) that $P(\beta_{x}=F)=\ell_{F}(x)/(n!\operatorname{vol}_{n}(\Gamma))$ . Hence,

	$\displaystyle{\mathcal{E}}(x)$	$\displaystyle=-\sum_{F}\frac{\ell_{F}(x)}{n!{\operatorname{vol}_{n}}(\Gamma)}\log\Big(\frac{\ell_{F}(x)}{n!{\operatorname{vol}_{n}}(\Gamma)}\Big)$
		$\displaystyle=\frac{1}{n!{\operatorname{vol}_{n}}(\Gamma)}\bigg(-\sum_{F}{\ell_{F}(x)}\log({\ell_{F}(x)})-\log({n!{\operatorname{vol}_{n}}(\Gamma)})\Big(\sum_{F}\lambda_{F}\Big)\bigg)$
		$\displaystyle=\frac{1}{n!{\operatorname{vol}_{n}}(\Gamma)}\bigg(\frac{\vartheta(x)}{c}-\log({n!{\operatorname{vol}_{n}}(\Gamma)})\Big(\sum_{F}\lambda_{F}\Big)\bigg).$

The result then follows from Theorem 6.37. ∎

Example 7.35.

The Fubini-Study metric of ${\mathcal{O}}(1)^{{\text{\rm an}}}$ corresponds to the case when $\Gamma$ and $\Delta$ are the standard simplex $\Delta^{n}$ and $c=1/2$ . In that case, the average entropy of the random variable $\beta_{x}$ is

\frac{1}{n!}\int_{\Delta^{n}}{\mathcal{E}}(x)\,\text{\rm d}\operatorname{vol}_{n}=\frac{2\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(\mathbb{P}^{n})}{(n+1)}=\sum_{j=2}^{n+1}\frac{1}{j}.

Remark 7.36.

In case $\Delta$ is a Delzant polytope whose facets have lattice volume 1, $\Gamma=\Delta$ , and $c=1/2$ , the roof function $\vartheta$ coincides with the symplectic potential of Guillemin canonical Kähler metric, see Remark 7.26.