From the definition it is easy to see that $\mathfrak{X}_{h^{\vee}}(q+t\mathfrak{X}_{h^{\vee}}(q))=\mathfrak{X}_{h^{\vee}}(q)$, for $q\in\partial\Delta^{\vee}_{\lambda}$, which immediately implies the straight ray foliation.

Note that as ${h^{\vee}}\to 0$, $L_{\nu,{h^{\vee}}}$ converges to $L_{\nu}$ uniformly in $(\mathbb{R}^{d})^{*}$. For $h^{\vee}=0$ the (discontinuous) vector field $\mathfrak{X}$ has (discrete) values in $\operatorname{vert}(T)$, and (1) and (2) follow immediately from the combinatorics of $\Sigma$. They remain true after regularization as well, which is guaranteed by the Proposition 3.1.

The bound (3) on the derivatives of $\mathfrak{X}$ follows from a standard estimate for regularization of piece-wise smooth function $L_{\nu}$. In the dual affine coordinates $\operatorname{Hess}L_{\nu}$ is a Dirac $\delta$-like distribution supported on $\partial\mathcal{V}$. The norm of its convolution with $\rho$ is bounded by $\frac{C}{(h^{\vee})^{k}}$, where $k$ is the codimension of the support. The constant $C$ takes into account the combinatorics of the polytope $\Delta$, the particular form of the mollifier $\rho$ and the choice of the norm on $\mathbb{R}^{d-1}\cong T_{q}({\Sigma\backslash D})$. The metric $g_{ij}$ appears from the chain rule: $\nabla\mathfrak{X}_{h^{\vee}}=g\cdot\operatorname{Hess}L_{\nu,h^{\vee}}$.

Finally, the smoothness of the vector field $\mathfrak{X}_{h^{\vee}}$, and hence the smoothness of the foliation $\mathcal{F}$, follow from smoothness of the map $\phi\hat{\phi}^{-1}$ on ${\Sigma\backslash D}$. ∎