Consider $h=0$ first. For a simplex $\tau\in T$, let $F_{\tau}$ be the corresponding face of ${\Delta_{\nu}}$, and we set $R_{\tau}:=\phi\hat{\phi}^{-1}(F_{\tau})+\operatorname{cone}(\tau)$. Then the set $\cup_{\tau^{\prime}\geq\tau}R_{\tau^{\prime}}$ contains some translation $R_{\geq\tau}$ of $\cup_{\tau^{\prime}\geq\tau}\operatorname{cone}(\tau^{\prime})$.

On the other hand, $\Phi$ is the Legendre transform of $\hat{\Phi}|_{\Delta_{\nu}}$. Hence, if $\nabla\Phi(x)=y$ and $r$ is in the normal cone to ${\Delta_{\nu}}$ at $y\in{\Delta_{\nu}}$, then $\nabla\Phi(x+r)=\nabla\Phi(x)=y$. So we see that for $x\in R_{\tau}$ the gradient $\nabla\Phi(x)$ takes values in the face $F_{\tau}$ of ${\Delta_{\nu}}$ because $x$. In particular, the $\tau$-slopes of $\Phi$ are equal to $-\nu|_{\tau}$.

For $h>0$ the statement of the lemma follows from the case $h=0$ and the Proposition 3.1. The translated cones $R_{\geq\tau}$ become shifted into their interiors by some vectors of size $h$. ∎