Many years ago Danzer resolved an open question of Erds by constructing a convex 9-gon, each vertex of which has the same distance to three other vertices. In Danzer's example, the replicated distance is not the same for all vertices. The present paper shows that it can be the same when n is somewhat larger than 9. In particular, there are convex n-gons with the following property. The vertices are partitioned into sets A and B on opposite sides of a line such that each a ε A is distance 1 from three vertices in B and each b ε B is distance 1 from three vertices in A. The smallest n for which this is possible is n = 20.