Caratheodory's Theorem (Convex Geometry)

Caratheodory's Theorem states that if a point lies in the convex hull of a set of points in d-dimensional space, then that same point already lies in the convex hull of some subset of at most d plus one of the original points. Named for Constantin Caratheodory, it bounds how many points are ever needed to express any point of a convex hull as a combination of the original set, and is a foundational tool of convex geometry and linear programming. Other results are also named for Caratheodory in other branches of mathematics, including a measure-theory extension theorem and a complex-analysis boundary theorem, neither of which is minted on this atlas yet.