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The following result was proved by C.Carathéodory in 1907 [1] :
Carathéodory’s theorem: If a point 
of 
lies in the convex hull 
of a set 
, there is a subset 
of 
consisting of no more than 
points such that 
lies in the convex hull of 
.
In other words, 
lies in a r-simplex with vertices in 
, where 
.
Equivalently, any convex combination of points in 
is a convex combination of at most 
of them.
Example: If a point 
in the plane 
is contained in a convex hull of a set 
, then there are at most three points in 
that determine the set which convex hull contains 
.
![[Graphics:HTMLFiles/CaratheodoryTheorem_19.gif]](HTMLFiles/CaratheodoryTheorem_19.gif)
| [1] | Carathéodory, C. (1907). Über den Variabilitãtsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen. Math. Ann., 64, 95-115. |
Cite this web-page as:
Štefan Porubský: Carathéodory’s Convex Hull Theorem.