- Radon's theorem
In
geometry , Radon's theorem onconvex set s, named afterJohann Radon , states that any set of points in R"d" can be partitioned into two (disjoint) sets whoseconvex hull s intersect. A point in the intersection of these hulls is called a Radon point of the set.For example, in the case , the set, call it , would consist of four points. Depending on the set, it might be possible to partition , into a triple and a singleton, where the convex hull of the triple (a triangle) contains the singleton, or it would be possible to partition , into two pairs of points such that the line segments with these points as endpoints intersect. The latter situation would be the case if , consists of the vertices of a convex
quadrilateral .Proof
The proof of Radon's theorem is not too difficult. Suppose . Since any set of points in is affinely dependent, there exists a set of multipliers not all zero such that the system of equations
:
:
is satisfied. Fix some nonzero solution . Let be the subset of such that for all , and let be the subset of such that for all . The required partition of is and . One point in the intersection of the convex hull of and that of is
:
It is clear that this point is in the convex hull of and it is an easy consequence of the above equations satisfied by that this point is also in the convex hull of . This completes the proof.
Tverberg's theorem
A generalisation for partition into "r" sets was given in 1966 by
Helge Tverberg . It states that for:("d" + 1)("r" − 1) + 1
points in Euclidean "d"-space, there is a partition into "r" subsets (Tverberg partition) having convex hulls intersecting in at least one common point.
ee also
*
Helly's theorem
* Carathéodory's theorem.References
*J. Eckhoff, Helly, Radon, and Carathéodory type theorems, Handbook of convex geometry, Vol. A, B, 389-448, North-Holland, Amsterdam, 1993.
*J. Radon, Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten, Math. Ann. Vol. 83 (1921), 113--115.
*H. Tverberg, A generalization of Radon's theorem, J. Lond. Math. Soc. Vol. 41 (1966), 123-128.
*S. Hell, Tverberg-type theorems and the Fractional Helly property, Dissertation, TU Berlin 2006 ( [http://opus.kobv.de/tuberlin/volltexte/2006/1416/ Full text] )
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