Carathéodory's theorem (convex hull)

Carathéodory's theorem (convex hull) - brief summary

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Carathéodory's theorem is a theorem in convex geometry. It states that if a point x of Rd lies in the convex hull of a set P, then x can be written as the convex combination of at most d + 1 points in P. Namely, 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 an r-simplex with vertices in P, where . The smallest r that makes the last statement valid for each x in the convex hull of P is defined as the Carathéodory's number of P. Depending on the properties of P, upper bounds lower than the one provided by Carathéodory's theorem can be obtained. Note that P need not be itself convex. A consequence of this is that P′ can always be extremal in P, as non-extremal points can be removed from P without changing the m.

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