Polygons Inscribed in a Convex Figure

Abstract

The paper contains a survey of results about the possibility of inscribing 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 another one lies inside K.