Every centered convex body in R^n with only the origin as an interior lattice point and volume (n+1)^n/n! is unimodularly equivalent to the centered standard simplex.
On the volume of K-semistable Fano manifolds
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abstract
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.
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The equality case of Ehrhart's volume conjecture
Every centered convex body in R^n with only the origin as an interior lattice point and volume (n+1)^n/n! is unimodularly equivalent to the centered standard simplex.