ScalingStacks

Proof. This argument is essentially the same as [36, Thm 3.1]. We quote [36, Lem 3.2]:

Using Thm. 5.6, we can find inside the regular region of $X_{s}$ some geodesic ball $B_{g_{CY,s}}(p,r)$ of radius $r<1$, occupying a nontrivial portion of the total volume:

$$\frac{\text{Vol}(B_{g_{CY,s}}(p,r)))}{\text{Vol}(X_{s})}\geq\epsilon>0,$$

with $\epsilon$ independent of $s$. Now applying the Lemma to the rescaled CY metric $r^{-2}g_{CY,s}$,

$$\frac{\text{diam}(X_{s})-r}{4nr}\leq\frac{\text{Vol}(B_{g_{CY,s}}(p,2(\text{diam}(X_{s})+r)))}{\text{Vol}(B_{g_{CY,s}}(p,r))}\leq\frac{\text{Vol}(X_{s})}{\text{Vol}(B_{g_{CY,s}}(p,r))}\leq\epsilon^{-1},$$

so $\text{diam}(X_{s})\leq Cr\leq C$ as required. ∎