Mathematics, Department of


Date of this Version



The American Mathematical Monthly, Vol. 87, No. 4 (Apr., 1980), pp. 284-285


Copyright 1980 Mathematical Association of America


In [1] Guggenheimer states that a Jordan polygon has two principal vertices that are exposed points of its convex hull, and he refers to Meisters's paper [3]. Such a statement cannot be found in Meisters's paper, and in fact it is false. The polygon illustrated in the figure below provides a counterexample.

By a Jordan polygon P = V1 . . . VN is meant a simple closed polygonal plane curve with N sides Vl V2, V2 V3, . . . , VN-1, VN, VN V1, joining the N vertices V1, . . . ,VN. In [3] any consecutive vertices Vi-1, Vi, and Vi+1 of a Jordan polygon P are said to form an ear (regarded as the region enclosed by the triangle Vi-1 Vi Vi-1) at the vertex Vi if the open chord joining Vi-1 andVi+1 lies entirely inside the polygon P. Two such ears are called nonoverlapping if the interiors of their triangular regions are disjoint. The following Two-Ears Theorem was proved in [3].

TWO-EARS THEOREM. Except for triangles, every Jordan polygon has at least two nonoverlapping ears.