  It is shown that that the rank of the second fundamental form (resp. the Levi
form) of a $\mathcal C^2$-smooth convex hypersurface $M$ in $\Bbb R^{n+1}$
(resp. $\Bbb C^{n+1}$) does not exceed an integer constant $k<n$ near a point
$p\in M,$ then through any point $q\in M$ near $p$ there exists a real (resp.
complex) $(n-k)$-dimensional plane that locally lies on $M.$
