Five theorems on polygons and polytopes inscribed in (or circumscribed about) a convex compact set in the plane or space are proved by topological methods. In particular, it is proved that for every interior point $O$ of a convex compact set in $\mathbb R^3$, there exists a two-dimensional section through $O$ circumscribed about an affine image of a regular octagon. It is also proved that every compact convex set in $\mathbb R^3$ (except the cases listed below) is circumscribed about an affine image of a cube-octahedron (the convex hull of the midpoints of the edges of a cube). Possible exceptions are provided by the bodies containing a parallelogram $P$ and contained in a cylinder with directrix $P$. Bibl. 29 titles.