Take a holomorphic maximally degenerating family of principally polarized abelian varieties $\pi\colon(\mathcal{X},\mathcal{L})\to\Delta$. Consider the rescaled Gromov-Hausdorff limit $B(\mathcal{X},\mathcal{L})$ of diameter $1$ as in Theorem 1.1 ([Od16]) and its discrete Legendre transform $\check{B}(\mathcal{X},\mathcal{L})$ ([GS11], [KS04]).

Then we can enhance the underlying integral affine structure of $\check{B}(\mathcal{X},\mathcal{L})$ as $K$-affine structure (in the sense of [KS04, §7.1]) naturally via the data of $\pi$. Furthermore, such $K$-affine structure recovers $\pi$ up to an equivalence relation generated by base change (replace $t$ by $t^{a}$ with $a\in\mathbb{Q}_{>0}$).