  Let $X$ be a Gorenstein minimal $3$-fold of general type. We prove the
optimal inequality: $$K_X^{3}\geq \frac{4}{3}\chi(\omega_X)-2,$$ where
$\chi(\omega_X)$ is the Euler-Poincar$\acute{\text{e}}$ characteristic of the
dualizing sheaf $\o_{X}$.
