I was recently wondering if there existed an algebraic definition for the concept of a convex polygon or its vertices. After searching on the Internet for some time, I couldn't find one, but I came up with the following definition myself: a point in a convex polygon is a vertex if it can't be expressed as a convex combination of any other points of that polygon.
Intuitively, it makes sense, but I don't know how to prove that. I guess I should first assume that a convex polygon is always the convex hull of a set of points, but I'm not quite sure of that either.