The H-representation of $\operatorname{c}(\psi)$ is

	$\displaystyle{\operatorname{dom}}(\operatorname{c}(\psi))$	$\displaystyle=\{(u,r)\in{\widetilde{N}}_{\mathbb{R}}\mid r\geq 0\},$
	$\displaystyle\operatorname{c}(\psi)(u,r)$	$\displaystyle=\min_{\Lambda}(m_{\Lambda}(u)+l_{\Lambda}r).$

By Proposition 3.64

$$\operatorname{stab}(\operatorname{c}(\psi))=\mathbb{R}_{\geq 0}(0,1)+\operatorname{conv}(\{(m_{\Lambda},l_{\Lambda})\}_{\Lambda\in\Pi}).$$

Furthermore, by the same proposition, for $x\in\operatorname{stab}(\psi)$,

$$\psi^{\vee}(x)=\sup\left\{\sum_{\Lambda}-\lambda_{\Lambda}l_{\Lambda}\bigg|\lambda_{\Lambda}\geq 0,\sum_{\Lambda}\lambda_{\Lambda}=1,\sum_{\Lambda}\lambda_{\Lambda}m_{\Lambda}=x\right\}.$$

Hence $\operatorname{epi}(-\psi^{\vee})=\mathbb{R}_{\geq 0}(0,1)+\operatorname{conv}(\{(m_{\Lambda},l_{\Lambda})\}_{\Lambda\in\Pi}),$ which proves the statement. ∎