Minimum-Weight Steiner Triangulation of Convex Polygons Requires Interior Steiner Points (opens in new tab)

We construct a convex polygon for which the minimum-weight Steiner triangulation requires an interior Steiner point. This provides a counterexample to a 1994 conjecture of Eppstein that minimum-weight Steiner triangulation of convex polygons needs only Steiner points on the boundary of the polygon.