Convex curves whose points are vertices of billiard triangles (Q1103872)

On a smooth convex curve of constant width all points are vertices of periodic orbits of billiard balls with period two. On ellipses every point is a vertex of a billiard n-gon for all \(n\geq 3\). The author shows that the converse is not true if the assumption is restricted to \(n=3\). More precisely, smooth convex curves are constructed which are different from an ellipse such that all points are vertices of billiard triangles, i.e. vertices of orbits of period three.