On the volume of K-semistable Fano manifolds

We prove that the anti-canonical volume of an $n$-dimensional K-semistable Fano manifold that is not ${\mathbb{P}}^n$ is at most $2n^n$. Moreover, the volume is equal to $2n^n$ if and only if $X\cong {\mathbb{P}}^1\times {\mathbb{P}}^{n-1}$ or $X$ is a smooth quadric hypersurface $Q\subset {\mathbb{P}}^{n+1}$. Our proof is based on a new connection between K-semistability and minimal rational curves.