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Carathéodory's theorem is a theorem in convex geometry. It states that if a point lies in the convex hull of a set , then lies in some -dimensional simplex with vertices in . Equivalently, can be written as the convex combination of or fewer points in . Additionally, can be written as the convex combination of at most extremal points in , as non-extremal points can be removed from without changing the membership of in the convex hull.
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