  Given a lattice polytope $Q\subset \mathbb{R}^n$, we can consider the cone
$\sigma=C(Q)=\{\lambda(q,1)\in \mathbb{R}^{n+1}|\lambda \in \mathbb{R}_{\geq0},
q\in Q\} \subset \mathbb{R}^{n+1}$, and the affine toric variety $Y_{\sigma}$
associated to $\sigma$. Altmann showed that the versal deformation space of
$Y_\sigma$ can be described by the Minkowski decomposition of the polytope $Q$.
Under some conditions on $Q$, we can obtain a smooth deformation $Y_\epsilon$
of $Y_\sigma$ using Altmann's result. In this article, we consider $Y_\epsilon$
inside some $\mathbb{C}^N$ and construct a complex fibration on $Y_\epsilon$,
with general fibre $(\mathbb{C}^*)^n$ and finite singular fibres described
using global coordinates related to the components of the Minkowski
decomposition. We construct a singular Lagrangian torus fibration out of the
complex fibration. This singular fibration admits a convex base diagram
representation with cuts as a natural generalization of base diagrams described
by Symington for Almost Toric Fibrations ($\dim=4$). In particular, we obtain a
convex base diagram whose image is the dual cone of $C(Q)$. There is a
1-parameter family of monotone Lagrangian tori in each of these fibrations.
Using the wall-crossing formula, we describe the potential associated with this
family in terms of the Minkowski decomposition of $Q$, recovering the result of
Lau, and discuss non-displaceability. We also discuss some other consequences
of our results.
