On common sections with given properties for finite set of convex compacts

The general theorem below is proved.
Theorem. {\it Let $O$ be the interior point for $n-2$ convex compacts $K_1,\dots,K_{n-2}$ in $\mathbb R^n$. There exists such two-dimensional plane $H$, passing through the point $O$, that for $i\le n-2$ some affine image of the given centrally-summetric hexagon is inscribed in $K_i\cap H$ and has the center at point $O$. There exist such $n-3$ two-dimensional planes $H_1,\dots,H_{n-3}$, passing through the point $O$, and laying at the same time in three-dimensional plane, that for $i\le n-3$ some affine image of regular octagon us inscribed in $H_i\cap K_i$ and has the center at point $O$.}