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Carathéodory's theorem (Theorem)

Suppose a point $ p$ lies in the convex hull of a set $ P\subset \mathbb{R}^d$. Then there is a subset $ P'\subset P$ consisting of no more than $ d+1$ points such that $ p$ lies in the convex hull of $ P'$.

For example, if a point $ p$ is contained in a convex hull of a set $ P\subset \mathbb{R}^2$, then there are three points in $ P$ that determine the triangle containing $ p$, provided, of course, that $ P$ contains at least three points.



"Carathéodory's theorem" is owned by bbukh.
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See Also: convex set


Attachments:
proof of Carathéodory's theorem (Proof) by kshum
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Cross-references: contains, triangle, contained, subset, convex hull, point
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This is version 3 of Carathéodory's theorem, born on 2003-09-12, modified 2004-12-24.
Object id is 4730, canonical name is CaratheodorysTheorem2.
Accessed 6267 times total.

Classification:
AMS MSC: 52A20 (Convex and discrete geometry :: General convexity :: Convex sets in $n$ dimensions )
Pending Errata and Addenda
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