Let $u$ be a continuous function on $\partial\Delta^{\vee}_{\lambda}$, to which we associate a collection of local functions $\{u_{m}\}_{m\in\Delta_{\mathbb{Z}}}$ by the rule $u_{m}=u-\langle x,m\rangle$. We regard $u_{m}$ as a function on the charts $\partial\Delta_{\lambda}^{\vee}\cap U_{w}^{\infty}$ with $\langle w,m\rangle=1$. We say $u$ is a locally convex function if all $u_{m}$ are convex on their corresponding charts.