Optimal bounds for the volumes of Kähler-Einstein Fano manifolds

We show that any $n$-dimensional Ding semistable Fano manifold $X$ satisfies that the anti-canonical volume is less than or equal to the value $(n+1)^n$. Moreover, the equality holds if and only if $X$ is isomorphic to the $n$-dimensional projective space. Together with a result of Berman, we get the optimal upper bound for the anti-canonical volumes of $n$-dimensional K\"ahler-Einstein Fano manifolds.