We prove that if a Q-Fano variety V specially degenerates to a Kähler-Einstein Q-Fano variety V, then for any ample Cartier divisor H=−r−1KV
Nearby in the stack
with
r∈Q>0
, the normalized volume
vol(v)=ACn(v)⋅vol(v)
is globally minimized at the canonical valuation
ordV
among all real valuations which are centered at the vertex of the affine cone
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
vol
over the corresponding Kähler cone. These results strengthen the minimization result of Martelli-Sparks-Yau [Martelli et al 08].