We will construct a foliation of $\mathbb{R}^{d}\backslash\Delta^{\vee}_{\lambda}$ by straight lines/rays by specifying a “convex” vector field on $\partial\Delta^{\vee}_{\lambda}$, which is smooth in $\Sigma^{\mathrm{sm}}$.

Recall that the piece-wise linear functions $L_{\nu}:(\mathbb{R}^{d})^{*}\to\mathbb{R}$ and $L_{\lambda}:\mathbb{R}^{d}\to\mathbb{R}$, the Legendre transforms of $\nu$ and $\lambda$, were defined as

$$L_{\nu}(m):=\max_{n\in\Delta^{\vee}_{\mathbb{Z}}}\{\langle m,n\rangle+\nu(n)\},\quad L_{\lambda}(n):=\max_{m\in\Delta_{\mathbb{Z}}}\{\langle m,n\rangle+\lambda(m)\}.$$

We fix a mollifier $\rho^{\vee}$ on $(\mathbb{R}^{d})^{*}$ with support in $\Delta$ and consider smooth functions $L_{\nu,h^{\vee}}:=\rho^{\vee}_{h^{\vee}}\ast L_{\nu-h^{\vee}}$. From the properties of regularization (for ${h^{\vee}}$ small in the $\nu$-scale) the slopes $\nabla L_{\nu,{h^{\vee}}}(x)$ always lie in $\partial\Delta^{\vee}$, for any $x$ not in the interior of ${\Delta_{\nu}}$. In particular, the gradient of $L_{\nu,{h^{\vee}}}$ gives a map $\nabla L_{\nu,{h^{\vee}}}:\partial{\Delta_{\nu}}\to\partial\Delta^{\vee}$.

Now we can use the identification of $\partial{\Delta_{\nu}}$ with $\partial\Delta^{\vee}_{\lambda}$ via $\phi\hat{\phi}^{-1}$ given by the $(\lambda,\nu)$-bi-PIKAS to define a $\partial\Delta^{\vee}$-valued vector field $\mathfrak{X}_{h^{\vee}}$ on $\partial\Delta^{\vee}_{\lambda}$ by

$$\mathfrak{X}_{h^{\vee}}(q):=\nabla L_{\nu,h^{\vee}}(\phi\hat{\phi}^{-1}q).$$

We can extend this vector field $\mathfrak{X}_{h^{\vee}}$ to $\mathbb{R}^{d}\backslash\Delta^{\vee}_{\lambda}$ using the following identification of $\partial{\Delta_{\nu}}$ with $\partial\Delta^{\vee}_{\lambda+t}$, for any $t\geq 0$. If $\hat{\Phi}$ is the (dual) $(\lambda,\nu)$-bi-PIKAS potential, then $\hat{\Phi}+tL_{\nu,{h^{\vee}}}$ is a strictly convex function on ${\Delta_{\nu}}$. In particular, its gradient defines a bijection $\nabla\hat{\Phi}+t\nabla L_{\nu,{h^{\vee}}}:\partial{\Delta_{\nu}}\to\partial\Delta^{\vee}_{\lambda+t}$. We can write the vector field $\mathfrak{X}_{h^{\vee}}$ on $\mathbb{R}^{d}\backslash\Delta^{\vee}_{\lambda}$ explicitly, using the fact that

$$L_{\lambda}(x)=t\Leftrightarrow x\in\partial\Delta^{\vee}_{\lambda+t},\text{ for any }t\geq 0.$$

Namely,

$$\mathfrak{X}_{h^{\vee}}(x):=\nabla L_{\nu,h^{\vee}}([\nabla\hat{\Phi}+L_{\lambda}(x)\nabla L_{\nu,{h^{\vee}}}]^{-1}(x)).$$

We summarize properties of this vector field in the following lemma: