Editing Carathéodory's theorem (convex hull) (section)

{{short description|Point in the convex hull of a set P in Rd, is the convex combination of d+1 points in P}}
{{other uses|Carathéodory's theorem (disambiguation)}}
'''Carathéodory's theorem''' is a theorem in [[convex geometry]]. It states that if a point <math>x</math> lies in the [[convex hull]] <math>\mathrm{Conv}(P)</math> of a set <math>P\subset \R^d</math>, then <math>x</math> lies in some ''<math>d</math>''-dimensional [[simplex]] with vertices in <math>P</math>. Equivalently, <math>x</math> can be written as the [[convex combination]] of <math>d+1</math> or fewer points in <math>P</math>. Additionally, <math>x</math> can be written as the convex combination of at most <math>d+1</math> ''extremal'' points in <math>P</math>, as non-extremal points can be removed from <math>P</math> without changing the membership of ''<math>x</math>'' in the convex hull.

An equivalent theorem for [[Conical combination|conical combinations]] states that if a point <math>x</math> lies in the [[conical hull]] <math>\mathrm{Cone}(P)</math> of a set <math>P\subset \R^d</math>, then <math>x</math> can be written as the conical combination of at most <math>d</math> points in <math>P</math>.<ref name="lp">{{Cite Lovasz Plummer|mode=cs1}}</ref>{{rp|257}} 

Two other theorems of [[Helly's theorem|Helly]] and [[Radon's theorem|Radon]] are closely related to Carathéodory's theorem: the latter theorem can be used to prove the former theorems and vice versa.<ref>{{cite conference |last1=Danzer |first1=L. |last2=Grünbaum |first2=B. |author2-link=Branko Grünbaum |last3=Klee |first3=V. |author3-link=Victor Klee |year=1963 |title=Convexity |url= |series=Proc. Symp. Pure Math. |publisher=[[American Mathematical Society]] |volume=7 |pages=101–179 |contribution=Helly's theorem and its relatives}} See in particular p.109</ref>

The result is named for [[Constantin Carathéodory]], who proved the theorem in 1911 for the case when <math>P</math> is [[Compact space|compact]].<ref>{{Cite journal|last=Carathéodory|first=C.|title=Über den Variabilitätsbereich der Fourier'schen Konstanten von positiven harmonischen Funktionen|journal=Rendiconti del Circolo Matematico di Palermo (1884–1940)|language=de|volume=32|issue=1|pages=193–217[see p.200 bottom]|doi=10.1007/BF03014795|year=1911|s2cid=120032616|url=https://link.springer.com/article/10.1007%2FBF03014795}}</ref> In 1914 [[Ernst Steinitz]] expanded Carathéodory's theorem for arbitrary sets.<ref>{{cite journal
|last=Steinitz
|first=Ernst
|author-link=Ernst Steinitz
|title=Bedingt konvergente Reihen und konvexe Systeme
|journal=J. Reine Angew. Math.
|volume=1913
|year=1913
|pages=128–175
|doi=10.1515/crll.1913.143.128
|issue=143
|s2cid=120411668
}}</ref>
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