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

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

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 $x\in\Delta^{\circ}$,

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.

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 $C_{k}(\Delta,u_{i},V)$.

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,

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

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 $\sum_{F}\ell_{F}=-\sum_{F}\lambda_{F}$.

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 ${\overline{L}}$.