Real and complex k-planes in convex hypersurfaces
  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 $M.$ 
Nikolai Nikolov
math.CV math.DG