Editing Radon's theorem (section)

==Applications==
The Radon point of any four points in the plane is their [[geometric median]], the point that minimizes the sum of distances to the other points.<ref>{{citation|title=Shortest Connectivity: An Introduction with Applications in Phylogeny|volume=17|series=Combinatorial Optimization|first=Dietmar|last=Cieslik|publisher=Springer|year=2006|isbn=9780387235394|page=6|url=https://books.google.com/books?id=4E0r3oWkn6AC&pg=PA6}}.</ref><ref>{{citation|title=Four-point Fermat location problems revisited. New proofs and extensions of old results|first=Frank|last=Plastria|authorlink=Frank Plastria|year=2006|doi=10.1093/imaman/dpl007|journal=IMA Journal of Management Mathematics|url=http://mosi.vub.ac.be/papers/Plastria2005_Fegnano.pdf|zbl=1126.90046|volume=17|issue=4|pages=387–396}}.</ref>

Radon's theorem forms a key step of a standard proof of [[Helly's theorem]] on intersections of convex sets;<ref>{{harvtxt|Matoušek|2002}}, p. 11.</ref> this proof was the motivation for Radon's original discovery of Radon's theorem.

Radon's theorem can also be used to calculate the [[VC dimension]] of ''d''-dimensional points with respect to linear separations. There exist sets of ''d''&nbsp;+&nbsp;1 points (for instance, the points of a regular simplex) such that every two nonempty subsets can be separated from each other by a hyperplane. However, no matter which set of ''d''&nbsp;+&nbsp;2 points is given, the two subsets of a Radon partition cannot be linearly separated. Therefore, the VC dimension of this system is exactly ''d''&nbsp;+&nbsp;1.<ref>[https://web.archive.org/web/20110722030339/http://ilex.iit.cnr.it/pellegrini/DispenseCorsoRandAlgPisa04/lez9-epsilonnet-draft.ps Epsilon-nets and VC-dimension], Lecture Notes by Marco Pellegrini, 2004.</ref>

A [[randomized algorithm]] that repeatedly replaces sets of ''d''&nbsp;+&nbsp;2 points by their Radon point can be used to compute an [[approximation algorithm|approximation]] to a [[Centerpoint (geometry)|centerpoint]] of any point set, in an amount of time that is polynomial in both the number of points and the dimension.<ref name="cemst"/>