  For a polarized abelian variety $(X,L)$, Z. Jiang and G. Pareschi introduce
an invariant $\beta(X,L)$, called the basepoint-freeness threshold. Using this
invariant, we show that a general polarized abelian variety $(X,L)$ of
dimension $g$ is projectively normal if $\chi(L) \geq 2^{2g-1}$ and the type of
$L$ is not $(2,4,\dots,4)$. This bound is sharp since it is known that any
polarized abelian variety of type $(2,4,\dots,4)$ is not projectively normal.
We also give an application of $\beta(X,L)$ to the Infinitesimal Torelli
Theorem for $Y \in |L|$.
