Since $\Psi$ is strictly concave on $\Sigma$, the Legendre-Fenchel correspondence shows that the set of rays of the form $\tau_{F}$ agrees with the set $\Sigma^{1}$. Moreover, $\Psi(v_{F})=\langle v_{F},F\rangle$, because $\Psi$ is the support function of $\Delta$. The proposition then follows from (4.25). ∎