The integral closedness of lattice simplices with large lattice length

We prove that every $n$-dimensional lattice simplex $P$ whose lattice length $L(P)\ge n-1$ is integrally closed. As an application, we obtain a simple criterion for the projective normality of ample line bundles on $\mathbb{Q}$-factorial toric Fano varieties with Picard number one. We further obtain a refinement of this result in terms of the invariant ${\mathrm{\Gamma }}_P$.