ScalingStacks

Notice $\text{Int}(\Delta_{J})\subset\Delta_{\mathcal{X}}$ inherits a natural integral affine structures. Since the restriction of $\phi_{0}$ is convex on these faces by Prop. 3.5, its real MA measure makes sense, and by Prop. 3.7 it satisfies the real MA equation on $\text{Int}(\Delta_{J})$

$$\text{MA}_{\mathbb{R}}(\phi_{0})=\frac{(L^{n})}{n!}d\mu_{0}.$$

(8)

Then the regularity theory of real MA equation (cf. section 2.5) will apply, so we may view the comparison property as a regularity assumption on $\left\lVert\cdot\right\rVert_{CY}$. Some subtleties are discussed in [21, Appendix].