I'm digging through "Computational Geometry" chapter in "Introduction to Algorithms" and there is a notion of convex hull introduced, alongside with Graham Scan algorithm for finding it.
One of the definitions in the book for a convex hull is:
Convex hull of a set of 2D points is the smallest convex polygon containing all points in a set.
The material is all intuitively clear for me, but when I try to prove to myself some elementary properties of 2D convex hull, I fail. In particular, I'm trying to prove the following properties:
- All vertices of convex hull of a set of points form a subset of these points
- Given a set of points, a point with minimal y-coordinate, which also has a minimal x-coordinate is a vertex of convex hull of that set
I just feel my mathematical toolbox is not rich enough to give such proofs. Using just a definition, I cannot get too far, as it doesn't give me much and seems too high level for me (notion of convexity and minimal area seems complex). Is there some particular branch of mathematics I should study before trying to prove that formally? Or maybe all tools I need are already in my toolbox and I should just think harder? Any tips?