K-semistability is equivariant volume minimization

This is a continuation to the paper [arXiv:1511.08164] in which a problem of minimizing normalized volumes over Q-Gorenstein klt singularities was proposed. Here we consider its relation with K-semistability, which is an important concept in the study of K\"{a}hler-Einstein metrics on Fano varieties. In particular, we prove that for a Q-Fano variety V, the K-semistability of (V, -KV) is equivalent to the condition that the normalized volume is minimized at the canonical valuation ordV among all C^*-invariant valuations on the cone associated to any positive Cartier multiple of -KV. In this case, it's shown that ordV is the unique minimizer among all C^*-invariant quasi-monomial valuations. These results allow us to give characterizations of K-semistability by using equivariant volume minimization, and also by using inequalities involving divisorial valuations over V.