Editing Radon's theorem (section)

=== Proofs ===
The topological Radon theorem was originally proved by Ervin Bajmóczy and [[Imre Bárány]]<ref name=":0">{{Cite journal |last1=Bajmóczy |first1=E. G. |last2=Bárány |first2=I. |date=1979-09-01 |title=On a common generalization of Borsuk's and Radon's theorem |url=https://doi.org/10.1007/BF01896131 |journal=Acta Mathematica Academiae Scientiarum Hungaricae |language=en |volume=34 |issue=3 |pages=347–350 |doi=10.1007/BF01896131 |s2cid=12971298 |issn=1588-2632}}</ref> in the following way:

* Construct a continuous map <math>g</math> from <math>S^d</math> (the <math>d</math>-dimensional [[N-sphere|sphere]]) to <math>\Delta^{d+1}</math>, such that for every point <math>x</math> on the sphere, <math>g(x)</math> and <math>g(-x)</math> are on two disjoint faces of <math>\Delta^{d+1}</math>.
* Apply the [[Borsuk–Ulam theorem]] to the function <math>f\circ g</math>, which is a continuous function from <math>S^d</math> to <math>\mathbb{R}^d</math>. The theorem says that, for any such function, there exists some point <math>y</math> on <math>S^d</math>, such that <math>f(g(y)) = f(g(-y))</math>. 
* The points <math>g(y)</math> and <math>g(-y)</math> are on two disjoint faces of <math>\Delta^{d+1}</math>, and they are mapped by <math>f</math> to the same point of <math>\mathbb{R}^d</math>. This implies that the images of these two disjoint faces intersect.

Another proof was given by [[László Lovász]] and [[Alexander Schrijver]].<ref>{{Cite journal |last1=Lovász |first1=László |last2=Schrijver |first2=Alexander |date=1998 |title=A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs |url=https://www.ams.org/proc/1998-126-05/S0002-9939-98-04244-0/ |journal=Proceedings of the American Mathematical Society |language=en |volume=126 |issue=5 |pages=1275–1285 |doi=10.1090/S0002-9939-98-04244-0 |s2cid=7790459 |issn=0002-9939|doi-access=free }}</ref> A third proof was given by [[Jiří Matoušek (mathematician)|Jiří Matoušek]]:<ref name=":03">{{Cite Matousek 2007}}, Section 4.3</ref>{{Rp|location=115}}

* Let <math>K</math> be the simplex <math>\Delta^{d+1}</math>, and let <math>K^{*2}_{\Delta}</math> be the [[deleted join]] of <math>K</math> with itself. 
* The geometric realization of <math>K^{*2}_{\Delta}</math>  is homeomorphic to the sphere <math>S^{d+1}</math>, therefore, the [[Z2-index|Z<sub>2</sub>-index]] of <math>K^{*2}_{\Delta}</math> equals <math>d+1</math>.
* The topological Radon theorem follows from the following more general theorem.  For any simplicial complex <math>K</math>, if the Z<sub>2</sub>-index of <math>K^{*2}_{\Delta}</math> is larger than <math>d</math>, then for every continuous mapping from <math>\|K\|</math> to <math>\mathbb{R}^d</math>, the images of two disjoint faces of <math>K</math> intersect.