We prove that the anti-canonical volume of an $n$-dimensional $K$-semistable Fano manifold that is not the projective space $P^n$ is at most $2n^n$. Moreover, the volume is equal to $2n^n$ if and only if $X$ is the product $P^1\times P^{n-1}$ or the smooth quadric hypersurface $Q$ in $P^{n+1}$.

The proof is based on a new connection between $K$-semistability and minimal rational curves. This is based on a joint work with Minghao Miao.