High Quality Content by WIKIPEDIA articles! In convex geometry Caratheodory's theorem states that if a point x of Rd lies in the convex hull of a set P, there is a subset P? of P consisting of d+1 or fewer points such that x lies in the convex hull of P?. Equivalently, x lies in a r-simplex with vertices in P, where r leq d. The result is named for Constantin Caratheodory, who proved the theorem in 1911 for the case when P is compact. In 1914 Ernst Steinitz expanded Caratheodory's theorem for any sets P in Rd.

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