Helly theorem

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Helly's theorem for Euclidean 2-dimensional space: If all triples of a set of surfaces intersect, then the intersection of all surfaces of the set is not empty.

The Helly's theorem is a mathematical theorem that the Austrian mathematician Eduard Helly back. The sentence is assigned to the field of convex geometry . Here it is closely related to a number of other classical theorems. Its effect also extends into other areas of mathematics, such as discrete mathematics , where it became the starting point for the investigation of set systems with the so-called Helly property .

Formulation of the sentence

The Helly's theorem can be formulated as follows:

Given a natural number and a lot of system of convex subsets of a dimensional normed vector space over and thereby applies   . In the whole system of sets there are only finitely many subsets or each of these subsets is compact in .
Then:
Necessary and sufficient for the   subsets occurring in to have a point in common is the condition that each of these subsets have a point in common.
In other words: With regard to the intersections, the following applies   if and only for all .

In another formulation, Helly's sentence can also be expressed as follows:

Under the above conditions is then and only then the intersection     , though for a single finite with       the intersection     is.  

The general requirements mentioned above can even be weakened to the extent that only the following is required for the infinite case:

Each of these subsets is closed in and at least one of these subsets is compact .

Historical, evidence, related findings

The Austrian mathematician Johann Radon provided the first proof of Helly's theorem in 1921. He used a result that is known today as the Radon theorem. Eduard Helly, however, had already found the sentence in 1913 at the latest and Johann Radon only proved the sentence after Eduard Helly had pointed it out to him. Eduard Helly himself then published two of his own works, which give a different approach to this topic. Other evidence has been found by other authors. The Helly's theorem is also an important tool in the proof of other classical theorems of convex geometry , such as the set of Krasnoselskii or in Jung's Theorem .

Demarcation

There are in the Analysis another Helly's theorem , which as a selection Helly's theorem is known or in the English literature as Helly's selection theorem . This deals with the convergence of function sequences .

literature

Original work

  • Eduard Helly: About sets of convex bodies with common points . In: Jahrb. Deut. Math. Association . tape 32 , 1923, pp. 175-176 .
  • Eduard Helly: About systems of closed sets with common points . In: monthly Math. Band 37 , 1930, pp. 281-302 .
  • Johann Radon: sets of convex bodies that contain a common point . In: Math. Ann. tape 83 , 1921, pp. 113-115 .

Monographs

  • 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.
  • Kurt Leichtweiß: Convex sets . Springer-Verlag, Berlin et al. 1980, ISBN 3-540-09071-1 .
  • Frederick A. Valentine: Convex sets (=  BI university paperbacks . Volume 402 / 402a). Bibliographical Institute , Mannheim 1968.

Web links

Individual evidence

  1. ^ VL Klee: Convexity . 1963, p. 101 ff .
  2. T. Bonnesen, W. Fennel: Theory of convex bodies. Corrected reprint . 1974, p. 3 .
  3. ^ FA Valentine: Convex sets . 1968, p. 78 .
  4. A. Brøndsted: An introduction to convex polytopes . 1983, p. 18 .
  5. J. Radon: sets of convex bodies that contain a common point . 1921, p. 113 .
  6. ^ SR Lay: Convex sets and their applications . 1982, p. 47 .
  7. ^ E. Helly: On sets of convex bodies with common points . 1923.
  8. ^ E. Helly: About systems of closed sets with common points . 1930.
  9. ^ FA Valentine: Valentine . 1968, p. 78 ff .
  10. ^ SR Lay: Convex sets and their applications . 1982, p. 53 ff .
  11. K. Leichtweiß: Convex sets . 1980, p. 70 ff .