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

=== Carathéodory's number ===
For any nonempty <math>P\subset \R^d</math>, define its '''Carathéodory's number''' to be the smallest integer <math>r</math>, such that for any <math>x\in \mathrm{Conv}(P)</math>, there exists a representation of <math>x</math> as a convex sum of up to <math>r</math> elements in <math>P</math>.

Carathéodory's theorem simply states that any nonempty subset of <math>\R^d</math> has Carathéodory's number <math>\leq d+1</math>. This upper bound is not necessarily reached. For example, the unit sphere in <math>\R^d</math> has Carathéodory's number equal to 2, since any point inside the sphere is the convex sum of two points on the sphere.

With additional assumptions on <math>P\subset \R^d</math>, upper bounds strictly lower than <math>d+1</math> can be obtained.<ref>{{Cite journal |last1=Bárány |first1=Imre |last2=Karasev |first2=Roman |date=2012-07-20 |title=Notes About the Carathéodory Number |journal=Discrete & Computational Geometry |language=en |volume=48 |issue=3 |pages=783–792 |arxiv=1112.5942 |doi=10.1007/s00454-012-9439-z |issn=0179-5376 |s2cid=9090617}}</ref>