In case we are given a strictly concave support function $\Psi$ on a fan $\Sigma$, then $N_{\Sigma}$ is homeomorphic to the polytope $\Delta_{\Psi}$ introduced in §4.4. An homeomorphism is obtained as the composition of ${\operatorname{\mathbf{e}}}_{K}$ with the moment map $\mu\colon X_{\Sigma}(\mathbb{R}_{\geq 0})\rightarrow\Delta_{\Psi}$ induced by $\Psi$:

$$\begin{matrix}N_{\Sigma}&\mathrel{\mathop{\kern 0.0pt\longrightarrow}\limits^{{\operatorname{\mathbf{e}}}_{K}}}&X_{\Sigma}(\mathbb{R}_{\geq 0})&\mathrel{\mathop{\kern 0.0pt\longrightarrow}\limits^{\mu}}&\Delta_{\Psi}\\[5.69054pt] u&\longmapsto&{\operatorname{\mathbf{e}}}_{K}(u)&\longmapsto&\frac{\sum\exp(-\lambda_{K}\langle m,u\rangle)m}{\sum\exp(-\lambda_{K}\langle m,u\rangle)}\end{matrix}$$

where the sums in the last expression are over the elements $m\in M\cap\Delta_{\Psi}$.