Here we give corrected versions of Definitions 1.3.3, 1.3.4, and 1.3.5. Then we'll prove a simpler alternative definition of the interior of a convex polygon. Of course, this means that, given a convex polygon, we'll define a set of points and prove that it is identical with the set defined in Definition 1.3.5.

Corrected Definitions

Definition 1.3.3. Let P1,...,Pn, n>3. be n points of the plane such that no triple Pi,Pi+1,Pi+2, for i=1,...,n-2; Pn-1,Pn,P1; Pn,P1,P2 is a collinear triple of points. Further assume that the segments P1--P2, P2--P3,..., Pn-1--Pn, Pn--P1 either have no point in common or have only an endpoint in common. Then the polygon P1P2...Pn is defined as the union of the line segments P1--P2, P2--P3,..., Pn-1--Pn, Pn--P1, which are called its sides. The points P1,...,Pn are called vertices. A polygon of n sides is also called an n-gon.

Definition 1.3.4 A polygon P1P2...Pn of n sides is called convex if for each side Pa--Pb, the vertices other than Pa or Pb are on the same side of the line <-Pa-Pb->. We call this side Hab.

Definition 1.3.5 Let Pn = P1P2...Pn be a convex polygon. We define the interior of Pn to be the intersection of all the half-planes Hab given in the definition of convex polygon.


It is a worthwhile exercise to show that the following alternate definitions of convex polygon and interior of a convex polygon are equivalent to 1.3.4 and 1.3.5.

Definition 1.3.4.0 A polygon P1P2...Pn of n sides is called convex if for each non-collinear ordered triple Pi,Pi+1,Pi+2, for i=1,...,n-2; Pn-1,Pn,P1; Pn,P1,P2, the vertices not in the triple are in the interior of the angle defined by the ordered triple.

Definition 1.3.5.0 Let Pn = P1P2...Pn be a convex polygon. We define the interior of Pn to be the intersection of the interiors of all the angles given in the definition of convex polygon.


Len M. Smiley
Last modified: Mon Feb 5 13:16:10 AKST 2001