Editing Radon's theorem (section)

=={{Anchor|topological}}Topological Radon theorem==
An equivalent formulation of Radon's theorem is:<blockquote>If ƒ is any [[affine function]] from a (''d''&nbsp;+&nbsp;1)-dimensional [[simplex]] Δ<sup>d+1</sup> to '''[[real number|R]]'''<sup>d</sup>, then there are two disjoint faces of Δ<sup>d+1</sup> whose images under ƒ intersect.</blockquote>They are equivalent because any affine function on a simplex is uniquely determined by the images of its vertices. Formally, let ƒ be an [[affine function]] from Δ<sup>d+1</sup> to '''[[real number|R]]'''<sup>d</sup>. Let <math>v_1,v_2,\dots,v_{d+2}</math> be the vertices of Δ<sup>d+1</sup>, and let <math>x_1,x_2,\dots,x_{d+2}</math> be their images under ''ƒ''. By the original formulation, the <math>x_1,x_2,\dots,x_{d+2}</math> can be partitioned into two disjoint subsets, e.g. (''x<sub>i</sub>'')<sub>i in I</sub> and (''x<sub>j</sub>'')<sub>j in J,</sub> with overlapping convex hull. Because ''f'' is affine, the convex hull of (''x<sub>i</sub>'')<sub>i in I</sub> is the image of the face spanned by the vertices (''v<sub>i</sub>'')<sub>i in I</sub>, and similarly the convex hull of (''x<sub>j</sub>'')j <sub>in J</sub> is the image of the face spanned by the vertices (''v<sub>j</sub>'')j <sub>in j</sub>. These two faces are disjoint, and their images under ''f'' intersect - as claimed by the new formulation. 

The '''topological Radon theorem''' generalizes this formluation. It allows ''f'' to be any continuous function - not necessarily affine:<ref name=":0" /><blockquote>If ƒ is any [[continuous function]] from a (''d''&nbsp;+&nbsp;1)-dimensional [[simplex]] Δ<sup>d+1</sup> to '''[[real number|R]]'''<sup>d</sup>, then there are two disjoint faces of Δ<sup>d+1</sup> whose images under ƒ intersect.</blockquote>More generally, if ''K'' is any (''d''&nbsp;+&nbsp;1)-dimensional compact convex set, and ƒ is any continuous function from ''K'' to ''d''-dimensional space, then there exists a linear function ''g'' such that some point where ''g'' achieves its maximum value and some other point where ''g'' achieves its minimum value are mapped by ƒ to the same point. In the case where ''K'' is a simplex, the two simplex faces formed by the maximum and minimum points of ''g'' must then be two disjoint faces whose images have a nonempty intersection. This same general statement, when applied to a [[hypersphere]] instead of a simplex, gives the [[Borsuk–Ulam theorem]], that ƒ must map two opposite points of the sphere to the same point.<ref name=":0" />

=== 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.