Intersection of Convex Sets

Convex sets are interesting in the context of optimization, as they represent regions where we know we can find global minima. A convex region is a region where for every pair of points within the region, every point on the line that joins the points is also within the region.

Fun fact - the intersection of all convex sets containing a given subset A of Euclidean space, is called the convex hull of A, which is the smallest convex set containing A. This is the definition needed in the development of Paul Schatz’ Oloid, which can be defined as “the convex hull of a skeletal frame made by placing two linked congruent circles in perpendicular planes, so that the center of each circle lies on the edge of the other circle”. More on that anon.

In the meantime, let’s tackle a simple proof to illustrate how the convex hull develops from the intersection of convex sets.

Let $S_{1}$ and $S_{2}$ be convex sets. Show that $S_{1} \cap S_{2}$ is convex.

Proof. Consider any arbitrary point $P \in S_{1} \cap S_{2}$. Without loss of generality, we have from the definition of intersections that $P \in S_{1}$, and we know that $S_{1}$ is a convex set. Since P was chosen arbitrarily, we have that $S_{1} \cap S_{2}$ is also a convex set. $\blacksquare$