Define $\eta\uparrow\lambda$ if $\eta=\lambda$ or $\eta=s_\alpha\cdot\lambda<\lambda$ for some $\alpha\in\Phi^+$. More generally, $\eta\uparrow\lambda$ if $\eta=\lambda$ or $\eta=s_{\alpha_1}s_{\alpha_2}\cdots s_{\alpha_r}\cdot\lambda\uparrow s_{\alpha_2}\cdots s_{\alpha_r}\cdot\lambda \uparrow \cdots \uparrow s_{\alpha_r}\lambda\uparrow\lambda$ for some $\alpha_1, \alpha_2,\cdots, \alpha_r\in\Phi^+$.
Let $\lambda\in\mathfrak{h}^*$ and $X(\lambda)$={$\eta$$\in\mathfrak{h}^*$$:\eta\uparrow\lambda\}$.
I would like to ask whether there is any geometry about $X(\lambda)$ or the convex hull of $X(\lambda)$.