Convex Polygons

Let's take the first definition, about the convex hull of a finite set of points $X$ .

WLOG, let's assume that

$X=\{x_1,x_2,x_3,\ldots,x_n\}$

so that $\|X\|=n$ (or, equivalently, $X$ consists of $n$ points), and that its convex hull has the set of points

$V=\{x_{i_1},x_{i_2},x_{i_3},\ldots,x_{i_m}\}$

as vertices.

Note that $m\leq n \Rightarrow m \text{ is finite}$ , and that $\|V\|=m$ .

For the sake of simplicity, let's label the $m$ points of $V$ in anticlockwise ascending order: by this I mean to assign the index $i_1$ to a random vertice, then the index $i_2$ to the vertice that is adjacent to $x_{i_1}$ when following the perimeter of the convex hull anticlockwise, or, more specifically, so that the interior (the bounded region of the plane) of the convex hull keeps always on the left when going from the vertice $x_{i_k}$ to the "next" one (which will always be the vertice with index $i_{k+1}$ , apart at the "end" of the cycle, because the next vertice with respect to $x_{i_m}$ will be $x_{i_1}$ ).

Note: $k \in \{1,\ldots,m-1\}$ .

This way, we have given the edges of the convex hull $\big($ which are the line segments with endpoints $(x_{i_k},x_{i_{k+1}})$ plus the line segment with endpoints $(x_{i_m},x_{i_1})\big)$ an orientation .

Now let's consider the set of oriented lines passing through the $m$ oriented edges of the convex hull, preserving the orientation we previously defined. Clearly, each of them divides the plane into two closed half-planes (if including the oriented line as the boundary).

Let's now consider the set of all closed half-planes that lie to le left of the various oriented lines. Clearly, their intersection is bounded , and is equivalent to the convex hull of the initial set of points $X$ . And, because $m$ is finite, there is a finite number of them.

As we can see, we have gone from the first definition of a convex polygon to the second one, and they are clearly

$\therefore\boxed{\text{equivalent}}.$