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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
.
[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.