Tverberg's theorem

The set of Tverberg ( English Tverberg's theorem ) is a theorem , both the mathematical field of convex geometry as well as that of the topological combinatorics attributable and in one of the Norwegian mathematician Helge Tverberg submitted in 1966, working back. It represents a generalization of the well-known radon theorem and is the starting point for a large number of more far-reaching studies. Closely related to it is Bárány’s theorem , from which Tverberg’s theorem can be derived.

Formulation of the sentence

Illustration for n = 2 and r = 3. N = 7 points allow a decomposition of the specified type.

The sentence says:

Let two natural numbers and the natural number be given . Was passed in Euclidean space is a subset consisting of at least spatial points are made to. ${\displaystyle n\geq 1}$ ${\displaystyle r\geq 2}$ ${\displaystyle N=(r-1)\cdot (n+1)}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle A\subseteq \mathbb {R} ^{n}}$ ${\displaystyle N+1}$

Then:

There is a decomposition

${\displaystyle A=A_{1}{\dot {\cup }}A_{2}{\dot {\cup }}\,\dots {\dot {\cup }}A_{r}}$

in pairwise disjoint subsets such that in the intersection ${\displaystyle r}$ ${\displaystyle A_{1},\ldots ,A_{r}\subseteq A}$

${\displaystyle \operatorname {conv} \left(A_{1}\right)\cap \operatorname {conv} \left(A_{2}\right)\cap \,\dots \cap \operatorname {conv} \left(A_{r}\right)}$

the associated convex hulls at least one common point in space.

Remarks

• Tverberg's theorem was preceded by a corresponding conjecture by the English mathematician Bryan John Birch , which he established in a paper presented in 1959.
• The proposition is optimal in the sense that the proposition of the proposition is no longer valid for subsets with at most spatial points. ${\displaystyle N}$
• For one obtains the theorem of radon . ${\displaystyle r=2}$

literature

• Imre Bárány , Pablo Soberón: Tverberg's theorem is 50 years old: A survey. In: Bull. Amer. Math. Soc. (NS) . tape 55 , 2018, p. 459-492 ( MR3854075 ).
• Bryan J. Birch : On 3N points in a plane . In: Proceedings of the Cambridge Philosophical Society . tape 55 , 1959, pp. 289-293 ( MR0109315 ).
• WA Coppel : Foundations of Convex Geometry (=  Australian Mathematical Society Lecture Series . Volume 12 ). Cambridge University Press , Cambridge 1998, ISBN 0-521-63970-0 ( MR1629043 ).
• Mark Longueville : A Course in Topological Combinatorics (=  university text ). Springer-Verlag, New York, Heidelberg, Dordrecht, London 2013, ISBN 978-1-4419-7909-4 ( MR2976494 ).
• Jiří Matoušek : Lectures on Discrete Geometry (=  Graduate Texts in Mathematics . Volume 212 ). Springer-Verlag , New York, Berlin, Heidelberg 2002, ISBN 0-387-95373-6 ( MR1899299 ).
• H. Tverberg : A generalization of Radon's theorem . In: The Journal of the London Mathematical Society . tape 41 , 1966, pp. 123-128 ( MR0187147 ).

Individual evidence

1. ^ WA Coppel: Foundations of Convex Geometry. 1998, p. 68 ff.
2. ^ Mark Longueville: A Course in Topological Combinatorics. 2013, p. 106 ff.
3. ^ Jiří Matoušek: Lectures on Discrete Geometry. 2002, p. 200 ff.
4. ^ WA Coppel: Foundations of Convex Geometry. 1998, p. 69.
5. ^ ^{A b} Mark Longueville: A Course in Topological Combinatorics. 2013, p. 106.
6. ^ Jiří Matoušek: Lectures on Discrete Geometry. 2002, p. 200.
7. ^ WA Coppel: Foundations of Convex Geometry. 1998, p. 70.