ScalingStacks

In section 4.2 we produced a collection of local average functions $\bar{\phi}=\bar{\phi}_{m,w}$ on $U_{w}^{s}$ corresponding to various choices of $w$ and $m$ with $\langle m,w\rangle=1$. But the local coordinates $x^{m_{1}},\ldots x^{m_{n}}$ are naturally interpreted also as coordinates on $\partial\Delta_{\lambda}^{\vee}$ (cf. section 3.2), so $\bar{\phi}_{m,w}$ can be alternatively viewed as a collection of convex functions on the charts $U_{w}^{\infty}\cap\partial\Delta_{\lambda}^{\vee}$ of $\partial\Delta_{\lambda}^{\vee}$. (Notice these local functions are defined without the need to shrink the domain to $U^{\infty}_{w,\delta}$).

The intuition is that up to $C^{0}$-small error, the differences of these local functions agree with the cocycle $\{m-m^{\prime}\}$, or equivalently, up to some $C^{0}$-small fuzziness $\bar{\phi}_{m,w}+\langle m,x\rangle$ glue to a locally convex function on $\partial\Delta_{\lambda}^{\vee}$ in the sense of Definition 3.22. The more precise statement is