  We study the Wu metric on convex egg domains of the form \[ E_{2m} = \big\{ z
\in \mathbb{C}^n : \vert z_1 \vert^{2m} + \vert z_2 \vert^2 + \ldots + \vert
z_{n-1} \vert^2 + \vert z_n \vert^{2} <1 \big\} \] where $m \geq 1/2, m \neq
1$. The Wu metric is shown to be real analytic everywhere except on a lower
dimensional subvariety where it fails to be $C^2$-smooth. Overall however, the
Wu metric is shown to be continuous when $m=1/2$ and even $C^1$-smooth for each
$m>1/2$, and in all cases, a non-K\"ahler Hermitian metric with its holomorphic
curvature strongly negative in the sense of currents. This gives a natural
answer to a conjecture of S. Kobayashi and H. Wu for such $E_{2m}$.
