Convex domains with locally Levi-flat boundaries

It is shown that a domain in $\C^n$ 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 $\C^{n-1}.$ This result does not extend to the case where the Levi form has locally constant rank for some $k<n-1$.