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.

**Definition 1.3.3.** Let P_{1},...,P_{n}, n__>__3. be n points of the plane such that no triple P_{i},P_{i+1},P_{i+2}, for i=1,...,n-2; P_{n-1},P_{n},P_{1}; P_{n},P_{1},P_{2} is a collinear triple of points. Further assume that the segments P_{1}--P_{2}, P_{2}--P_{3},..., P_{n-1}--P_{n}, P_{n}--P_{1} either have no point in common or have only an endpoint in common. Then the **polygon** P_{1}P_{2}...P_{n} is defined as the union of the line segments P_{1}--P_{2}, P_{2}--P_{3},..., P_{n-1}--P_{n}, P_{n}--P_{1}, which are called its **sides**. The points P_{1},...,P_{n} are called **vertices**. A polygon of n sides is also called an **n-gon**.

**Definition 1.3.4** A polygon P_{1}P_{2}...P_{n} of n sides is called **convex** if for each side P_{a}--P_{b}, the vertices other than P_{a} or P_{b} are on the same side of the line <-P_{a}-P_{b}->. We call this side H_{ab}.

**Definition 1.3.5** Let **P**_{n} = P_{1}P_{2}...P_{n} be a convex polygon. We define the interior of **P**_{n} to be the intersection of all the half-planes H_{ab} 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 P_{1}P_{2}...P_{n} of n sides is called **convex** if for each non-collinear ordered triple P_{i},P_{i+1},P_{i+2}, for i=1,...,n-2; P_{n-1},P_{n},P_{1}; P_{n},P_{1},P_{2}, the vertices not in the triple are in the interior of the angle defined by the ordered triple.

**Definition 1.3.5.0** Let **P**_{n} = P_{1}P_{2}...P_{n} be a convex polygon. We define the **interior** of **P**_{n} 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