Faces in rectilinear drawings of complete graphs

We initiate the study of extremal problems about faces in convex rectilinear drawings of $K_n$, that is, drawings where vertices are represented by points in the plane in convex position and edges by line segments between the points representing the end-vertices. We show that if a convex rectilinear drawing of $K_n$ does not contain a common interior point of at least three edges, then there is always a face forming a convex 5-gon while there are such drawings without any face forming a convex $k$-gon with $k\ge 6$.

A convex rectilinear drawing of $K_n$ is regular if its vertices correspond to vertices of a regular convex $n$-gon. We characterize positive integers $n$ for which regular drawings of $K_n$ contain a face forming a convex 5-gon.

To our knowledge, this type of problems has not been considered in the literature before and so we also pose several new natural open problems.