The canonical extension $u$ satisfies an a priori Lipschitz bound

$$\begin{cases}|u-\max_{m}\langle m,x\rangle|\leq C,\quad\forall x\in N_{\mathbb{R}},\\ |u(x)-u(x^{\prime})|\leq C|x-x^{\prime}|,\quad\forall x,x^{\prime}\in N_{\mathbb{R}}.\end{cases}$$

(25)

Morever, in the region $\text{Star}(w)+\mathbb{R}_{\geq 0}w\subset N_{\mathbb{R}}$, for any $m$ with $\langle m,w\rangle=1$, the function $u_{m}=u-m$ is constant upon translation in the $w$-direction.