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