On polygons inscribed into a convex figure

The paper contains a survey of results about the possibility to inscribe convex polygons of particular types into a plane convex figure. It is proved that if $K$ is a smooth convex figure, then $K$ is circumscribed either about four different reflection-symmetric convex equilateral pentagons or about a regular pentagon.
Let $S$ be a family of convex hexagons whose vertices are the vertices of two negatively homothetic equilateral triangles with common center. It is proved that if $K$ is a smooth convex figure, then $K$ is circumscribed either about a hexagon in $S$ or about two pentagons with vertices at the vertices of two hexagons in $S$. In the latter case, the sixth vertex of one of the hexagons lies outside $K$, while the sixth vertex of anther one lies inside $K$.