Concept

Carathéodory's theorem (convex hull)

Summary
Carathéodory's theorem is a theorem in convex geometry. It states that if a point x lies in the convex hull \mathrm{Conv}(P) of a set P\subset \R^d, then x can be written as the convex combination of at most d+1 points in P. More sharply, x can be written as the convex combination of at most d+1 extremal points in P, as non-extremal points can be removed from P without changing the membership of x in the convex hull. Its equivalent theorem for conical combinations states that if a point x lies in the conical hull \mathrm{Cone}(P) of a set P\subset \R^d, then x can be written as the conical combination of at most d points in P. The similar theorems of Helly and Radon are closely related to Carathéodory's theorem: the latter theorem can be used to prove the former the
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