K\"ahler-Einstein metrics and volume minimization

We prove that if a $\mathbb{Q}$-Fano variety $V$ specially degenerates to a K"{a}hler-Einstein $\mathbb{Q}$-Fano variety $V$, then for any ample Cartier divisor $H=-r^{-1}{K}_V$ with $r\in {\mathbb{Q}}_{>0}$, the normalized volume $\widehat{\mathrm{v}\mathrm{o}\mathrm{l}}(v)=A_{\mathcal{C}}^n(v)\cdot \mathrm{v}\mathrm{o}\mathrm{l}(v)$ is globally minimized at the canonical valuation ${\mathrm{o}\mathrm{r}\mathrm{d}}_V$ among all real valuations which are centered at the vertex of the affine cone $\mathcal{C}:=C(V,H)$. This is also generalized to the logarithmic and the orbifold setting. As a consequence, we complete the confirmation of a conjecture in [arXiv:1511.08164] on an equivalent characterization of K-semistability for any smooth Fano manifold. We also prove that the valuation associated to the Reeb vector field of a smooth Sasaki-Einstein metric minimizes $\widehat{\mathrm{v}\mathrm{o}\mathrm{l}}$ over the corresponding K"ahler cone. These results strengthen the minimization result of Martelli-Sparks-Yau [Martelli et al 08].