This problem could be interesting on its own, but as it turns out the convex hull has many usages.
For example, in Math — Gauss–Lucas theorem states that the roots of a polynomial’s derivative all lie within the convex hull of the roots of the polynomial (experimenting with this one below). In economics theory — some models have assumptions regarding the convexity of the data; When the data is not convex, it can be made convex by taking the convex hull. In robotics — planning the optimal path that avoids a polygonal obstacle.
And let’s check out the Gauss–Lucas theorem for the polynomial P(x) =
x???-x???+3x³-x²-3. On the complex plane, in green we can see that the roots of the derivative of P are indeed within the convex shape formed by the roots of P.