ScalingStacks

Remark 4.13. The tropical version $U^{\infty}_{w}$ of $U^{s}_{w}$ is in general larger than $U^{\infty}_{w}\cap\partial\Delta_{\lambda}^{\vee}$; it typically contains also some subset stretching to infinity along the $w$-direction. If we regard $\bar{\phi}_{m,w}$ as local functions on $\mathcal{A}_{\lambda}^{\infty}$ instead of $\partial\Delta_{\lambda}^{\vee}$, then there is a delicate issue. The Lemma above does not imply that $\bar{\phi}_{m,w}+\langle m,x\rangle$ for various choices of $m,w$ glue approximately on overlapping regions far from $\partial\Delta_{\lambda}^{\vee}$. The problem is that such overlapping regions have too small measure, which breaks down the proof.