Exceptional directions for a convex body

Let $K$ be a convex body in Rn andO be a point inside $K$. We examine the Grassmann manifold of $k$-planes passing through $O$. We take as exceptional the planes intersecting $K$ along a body having at least one $(k-1)$-dimensional face such that it does not have points inside the hyperfaces of body $K$. We prove that in the Grassmann manifold $G_k^n$ the set of such exceptional planes is of measure zero.