Theorem of radon

The set of Radon (also known as lemma Radon referred to) is a theorem of the convex geometry , which is based on the Austrian mathematician Johann Radon back. The theorem is directly related to Helly's theorem and is linked to other classical theorems of convex geometry through it.

Formulation of the sentence

The sentence can be formulated in the modern version as follows:

A natural number and a one - dimensional , real vector space as well as a subset of , which should consist of at least elements, are given. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle V}$ ${\displaystyle A}$ ${\displaystyle V}$ ${\displaystyle n+2}$

Then:

${\displaystyle A}$can such two disjoint subsets and dismantled are that their convex hulls , and in at least one point cut . ${\displaystyle A_{1}}$ ${\displaystyle A_{2}}$ ${\displaystyle \operatorname {conv} A_{1}}$ ${\displaystyle \operatorname {conv} A_{2}}$

Historical

Johann Radon formulated and proved the theorem in 1921. He then derived Helly's theorem from it, which Eduard Helly had already found in 1913 and which Johann Radon had communicated later.

Demarcation

Another important theorem of mathematics goes back to Johann Radon, namely the theorem of Radon-Nikodým , which, however, is not assigned to convex geometry, but to measure theory .

generalization

The theorem of Radon was generalized by Helge Tverberg in 1966 to the theorem of Tverberg .

literature

Original work

• Johann Radon: sets of convex bodies that contain a common point . In: Math. Ann . tape 83 , 1921, pp. 113-115 .

Monographs

• Arne Brøndsted: An introduction to convex polytopes . Springer-Verlag, New York a. a. 1983, ISBN 0-387-90722-X .
• Peter M. Gruber : Convex and Discrete Geometry . Springer-Verlag, Berlin a. a. 2007, ISBN 978-3-540-71132-2 .
• Victor L. Klee (Ed.): Convexity. Proceedings of the Seventh Symposium in Pure Mathematics of the American Mathematical Society, held at the University of Washington, Seattle, Washington, June 13-15, 1961 . American Mathematical Society , Providence, RI 1963.
• Steven R. Lay: Convex sets and their applications . John Wiley & Sons, New York a. a. 1982, ISBN 0-471-09584-2 .

Web links

• Ivan Izmestiev: Introduction to Convex Geometry. (PDF; 548 kB) FU Berlin, WS 03/04 (script)
• Günter M. Ziegler : 3N colored dots in the plane (about Birch's conjecture and Tverberg's theorem) . (PDF; 302 kB)

Individual evidence

1. ↑ See first web link
2. ↑ Klee: p. 101 ff.
3. ^ Brøndsted: p. 15
4. ↑ Gruber: p. 46 ff.
5. ↑ Klee: p. 103
6. ^ Radon: p. 113
7. ↑ Klee: p. 101
8. ↑ Lay: p. 47
9. ↑ See also second web link!