Steiner minimal trees for regular polygons.

Fifty years ago Jarnik and Kossler showed that a Steiner minimal tree for the vertices of a regular n-gon contains Steiner points for 3 = n = 5 and contains no Steiner point for n = 6 and n >= 13. We complete the story by showing that the case for 7 = n = 12 is the same as n >= 13. We also show that the set of n equally spaced points yields the longest Steiner minimal tree among all sets of n cocircular points on a given circle.