On K-stability and the volume functions of ℚ-Fano varieties: Table 1.

We introduce a new effective stability named ‘divisorial stability’ for Fano manifolds that is weaker than K-stability and is stronger than slope stability along divisors. We show that we can test divisorial stability via the volume function. As a corollary, we prove that the first coordinate of the barycenter of the Okounkov body of the anticanonical divisor is not bigger than one for any Kähler–Einstein Fano manifold. In particular, for toric Fano manifolds, the existence of Kähler–Einstein metrics is equivalent to divisorial semistability. Moreover, we find many non-Kähler–Einstein Fano manifolds of dimension 3.