On approximation of the plane sections of convex bodies

Topological methods are applied to the proof of three theorems concerning approximation of plane sections of convex bodies by affine-regular polygons, ellipses, or circles. One of the theorems is as follows. For every interior point $O$ of any convex body $K\subset\mathbb R^3$ there is a plane section of $K$ that passes through $O$ and admit an inscribed affine-regular hexagon centered at $O$. For every interior point $O$ of any convex body $K\subset\mathbb R^4$ there is a two-dimensional plane section of $K$ that passes through $O$ and admits an inscribed affine-regular octagon centered at $O$.