ScalingStacks

Lemma 2.9. (Volume-capacity estimate) In the setting of Thm. 2.7, for any compact set $K\subset X$,

$$\frac{\int_{K}\omega_{\phi}^{n}}{\text{Vol}(X)}\leq Ae^{\alpha}\exp\left(-\alpha(\frac{\text{Vol}(X)}{\text{Cap}_{\omega}(K)})^{1/n}\right).$$

(3)

In particular there is a constant $B=B(n,\alpha,A)$ verifying the power law bound

$$\frac{\int_{K}\omega_{\phi}^{n}}{\text{Vol}(X)}\leq B^{2n}\left(\frac{\text{Cap}_{\omega}(K)}{\text{Vol}(X)}\right)^{2}.$$