The Journal of Online Mathematics and Its Applications, Volume 8 (2008)

The Most Marvelous Theorem in Mathematics, Dan Kalman

A convex set has this defining property: For any two points of the set, the line segment joining the points is contained in the set. In Figure 1 below, the colored region on the left is convex, but the one on the right is not. In the region on the right, the two red points show that the convexity property fails, because the line segment joining these points does not stay within the region.

Given a set `A` of points, the convex hull is the smallest convex set containing `A`. When `A` has only a finite number of points, the convex hull is a polygon whose vertices are elements of `A`, although there may be additional elements of `A` in the interior of this polygon. That is the case in Figure 2 below.

One way to think of the convex hull is as follows: stretch a rubber band to make a giant circle, so big that the entire set `A ` its inside. Now let the rubber band shrink until it just contains `A`. The outline of the rubber band marks the boundary of the convex hull. You can see this in the animated graphic below. Click on the green arrow below the graphic to run the animation.