Convex Geometry |

The most fundamental principle in convex geometry follows from the geometric Hahn-Banach theorem which guarantees any closed convex set to be an intersection of halfspaces. The second most fundamental principle of convex geometry also follows from the geometric Hahn-Banach theorem that guarantees existence of at least one hyperplane supporting a convex set (having nonempty interior) at each point on its boundary. The third most fundamental principle of convex geometry again follows from the geometric Hahn-Banach theorem that guarantees existence of a hyperplane separating two nonempty convex sets whose relative interiors are nonintersecting. Separation intuitively means each set belongs to a halfspace on an opposing side of the hyperplane. |