Lemma 3.4 (00Q7) · ScalingStacks

Verified tagged author-source HTML · 1912.02360v1 · cited publication edition alignment unverified.

Lemma 3.4. There is a fixed number $\delta_{1}>0$ , such that for $s\gg 1$ and any $x\in\mathcal{A}^{s}_{\lambda}$ (or $x\in\mathcal{A}^{\infty}_{\lambda}$ ), there is a simplex $\sigma$ in the triangulation of $\Delta$ , verifying $\langle x,m\rangle+\lambda(m)<L_{\lambda}(x)-\delta_{1}$ for $m\in\Delta_{\mathbb{Z}}\setminus\sigma$ .

00Q8

Proof. (Sketch) For any fixed $x\in N_{\mathbb{R}}$ , the function $\langle x,m\rangle+\lambda(m)$ is a concave function of $m\in M_{\mathbb{R}}$ . By our assumptions, the set of $m\in\Delta_{\mathbb{Z}}$ saturating the maximum must be the set of vertices of some simplex $\sigma$ in the triangulation of $\Delta$ . A more effective version of this observation is the Lemma in the $\mathcal{A}^{\infty}_{\lambda}$ case, and the $\mathcal{A}^{s}_{\lambda}$ case follows by Prop. 3.2. ∎