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Carathéodory's theorem
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(Theorem)
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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.
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"Carathéodory's theorem" is owned by bbukh.
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Cross-references: contains, triangle, contained, subset, convex hull, point
There is 1 reference to this entry.
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 8284 times total.
Classification:
| AMS MSC: | 52A20 (Convex and discrete geometry :: General convexity :: Convex sets in $n$ dimensions ) |
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Pending Errata and Addenda
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