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It is shown that that the rank of the second fundamental form (resp. the Leviform) 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 $M.$

Real and complex k-planes in convex hypersurfaces