Convex domains with locally Levi-flat boundaries

It is shown that a domain in ${\text{C}}^n$ which is locally convex and has $\mathcal C^1$-smooth Levi-flat boundary is locally linearly equivalent to a Cartesian product of a planar domain and ${\text{C}}^{n-1}\mathrm{.}$ This result does not extend to the case where the Levi form has locally constant rank for some $k<n-1$.