  Let $\Omega$ be a bounded convex domain in $\mathbb{C}^{n}$, $n\geq 2$,
$1\leq q\leq (n-1)$, and $\phi\in C(\bar{\Omega})$. If the Hankel operator
$H^{q-1}_{\phi}$ on $(0,q-1)$--forms with symbol $\phi$ is compact, then $\phi$
is holomorphic along $q$--dimensional analytic (actually, affine) varieties in
the boundary. We also prove a partial converse: if the boundary contains only
`finitely many' varieties, $1\leq q\leq n$, and $\phi\in C(\bar{\Omega})$ is
analytic along the ones of dimension $q$ (or higher), then $H^{q-1}_{\phi}$ is
compact.
