By the definition of adelic field, ${\operatorname{val}}_{\mathbb{K}_{v}}(p)=0$ for almost all $v\in\mathfrak{M}_{\mathbb{K}}$. Therefore, the integrability of $Y$ follows as in the proof of Proposition 6.35.

Let $\Sigma$ be the complete regular fan of $N_{\mathbb{R}}$ induced by $H$ and $\Sigma_{\Delta^{r}}$, and let $X_{\Sigma}$ be the associated toric variety. Write $\varphi=\varphi_{H,p}$ for short. The fact that $H(N)$ is saturated implies that $\varphi$ has degree 1 and so $Y=\varphi_{*}X_{\Sigma}$. By the functoriality of the global height (Theorem 2.57(2)),

$$\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}^{{\operatorname{can}}}}(Y)=\operatorname{h}_{\varphi^{*}({\overline{{\mathcal{O}}(1)}}^{{\operatorname{can}}})}(X_{\Sigma}).$$

Let $v\in\mathfrak{M}_{\mathbb{K}}$. Using the results in Example 6.31, it follows from Theorem 6.37 that

$$\operatorname{h}_{\varphi^{\ast}({\overline{{\mathcal{O}}}}(1)^{{\operatorname{can}}})}(X_{\Sigma})=\left[(n+1)!\sum_{v}\int_{{\overline{\Delta}}}n_{v}{\overline{\vartheta}}_{v}\,\text{\rm d}\operatorname{vol}_{M}\right].$$

where ${\overline{\Delta}}=\operatorname{conv}(0,m_{1}-m_{0},\dots,m_{r}-m_{0})\subset M_{\mathbb{R}}$ and ${\overline{\vartheta}}_{v}$ is the function parameterizing the upper envelope of the extended polytope

$$\operatorname{conv}\left((0,0),(m_{1}-m_{0},\log|p_{1}/p_{0}|_{v}),\dots,(m_{r}-m_{0},\log|p_{r}/p_{0}|_{v})\right)\subset M_{\mathbb{R}}\times\mathbb{R}.$$

We have that ${\overline{\Delta}}=\Delta-m_{0}$ and ${\overline{\vartheta}}_{v}=\tau_{-m_{0}}\vartheta_{v}-\log|p_{0}|_{v}$. Hence,

$$\int_{{\overline{\Delta}}}{\overline{\vartheta}}_{v}\,\text{\rm d}\operatorname{vol}_{M}=\int_{\Delta}\vartheta_{v}\,\text{\rm d}\operatorname{vol}_{M}-\log|p_{0}|_{v}\operatorname{vol}_{M}(\Delta).$$

Since $\sum_{v}n_{v}\log|p_{0}|_{v}\in\operatorname{def}(\mathbb{K}^{\times})$, we deduce the result. ∎