ScalingStacks

Definition 3.29. Let $u$ be a locally convex function on $\partial\Delta_{\lambda}^{\vee}$ invariant under the discrete symmetry. Then $u$ is called an Aleksandrov solution of the real MA equation on $\partial\Delta_{\lambda}^{\vee}\setminus Sing$ if

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  On the interior of any top dimensional face of $\partial\Delta_{\lambda}^{\vee}$, in a set of standard local affine coordinates $x^{m_{1}},\ldots x^{m_{n}}$ with $dx^{m_{1}}\wedge dx^{m_{2}}\ldots dx^{m_{n}}$ equal to the standard volume form $d\mu_{\infty}$, the function $u$ satisfies $MA(u)=d\mu_{\infty}$ in the Aleksandrov sense.

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  On $\text{Star}(w)\subset U_{w}^{\infty}\cap\partial\Delta_{\lambda}^{\vee}$, we use the standard affine coordinates $x^{m_{1}},\ldots x^{m_{n}}$ associated to the $U_{w}^{\infty}\cap\partial\Delta_{\lambda}^{\vee}$ chart. We demand for any vertex $m$ of $\Delta$ with $\langle m,w\rangle$, the local function $u_{m}=u-m$ satisfies $MA(u_{m})=d\mu_{\infty}$ in the Aleksandrov sense.

Schematically we write $MA(u)=d\mu_{\infty}$.