Corollary 21.13.61.E1B465B480AB There is a canonical bijection
\[ \mathrm{PSH}_{\mathrm{tor}}(X,\theta )/\sim _{P}=\mathrm{PSH}_{\mathrm{tor}}(X,\theta )/\sim _{\mathcal{I}}\rightarrow \left\{ K\in \mathcal{K}_ n: K \subseteq P_ D \right\} . \]
Here $\mathcal{K}_ n$ denotes the set of convex bodies in $\mathbb {R}^ n$, as in subsection 19.7.1.
Proof.
Thanks to corollary 21.13.60, we have the equality
\[ \mathrm{PSH}_{\mathrm{tor}}(X,\theta )/\sim _{P}=\mathrm{PSH}_{\mathrm{tor}}(X,\theta )/\sim _{\mathcal{I}}, \]
and the map
\[ \mathrm{PSH}_{\mathrm{tor}}(X,\theta )/\sim _{\mathcal{I}}\rightarrow \left\{ K\in \mathcal{K}_ n: K \subseteq P_ D \right\} \]
is well-defined and injective.
In order to see the surjectivity, take $K\in \mathcal{K}_ n$, $K\subseteq P_ D$. Let $\chi _ K$ be the characteristic function of $K$ as defined in example 21.11.5. Since $\chi _ K$ satisfies Condition 3 of theorem 21.13.53, we can find $\varphi \in \mathrm{PSH}_{\mathrm{tor}}(X,\theta )$ so that $\chi _ K=G_{\varphi }$. Then $\Delta (\theta ,\varphi )=K$ due to proposition 21.11.34.
$\square$
Pome
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