  We show that a general $n$-dimensional polarized abelian variety $(A,L)$ of a
given polarization type and satisfying $ h^0(A, L) \geq \dfrac{8^n}{2} \cdot
\dfrac{n^n}{n !}$ is projectively normal. In the process, we also obtain a
sharp lower bound for the volume of a purely one-dimensional complex analytic
subvariety in a geodesic tubular neighborhood of a subtorus of a compact
complex torus.
