Kirchberger's theorem

The set of Kirchberger is one of the classic tenets of the mathematical part of the area of the convex geometry . It goes back to the dissertation of the mathematician Paul Kirchberger and is closely related to and even a direct consequence of the well-known Helly theorem . The kirchberger theorem gave rise to further research and to the discovery of a number of theorems of a similar type.

Formulation of the sentence

The set of Kirchberger can be specified as follows:

Given a natural number , and two finite sets and thereby are for each of a maximum of points in space existing subset , the two subsets and always by a hyperplane of strictly separated . ${\displaystyle n}$ ${\displaystyle A,B\subset \mathbb {R} ^{n}}$ ${\displaystyle n+2}$ ${\displaystyle C\subseteq {A\cup B}}$ ${\displaystyle C\cap A}$ ${\displaystyle C\cap B}$ ${\displaystyle \mathbb {R} ^{n}}$

Then:

${\displaystyle A}$and are also strictly separable by a hyperplane of . ${\displaystyle B}$ ${\displaystyle \mathbb {R} ^{n}}$

extension

Kirchberger's theorem can be extended by weakening the assumption that the point sets are finite . The assertion of the theorem remains also in the event that - other things being equal - and only as compact subsets of is assumed. This extended sentence is also referred to as the Kirchberger sentence. ${\displaystyle A,B\subset \mathbb {R} ^{n}}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \mathbb {R} ^{n}}$

To the history

Paul Kirchberger was a student of David Hilbert and has in this in 1902 with the dissertation Tschebyschefsche About approximation methods doctorate . Kirchberger published extracts from this dissertation in volume 57 of the Mathematische Annalen in 1903. The sentence presented here appears there in Chapter III ("An auxiliary sentence"). As some authors - such as Alexander Barvinok and Steven R. Lay - point out, Kirchberger proved his theorem several years before the publication (and therefore without the help of) of Helly's theorem.

literature

• Alexander Barvinok: A Course in Convexity (=  Graduate Studies in Mathematics . Volume 54 ). American Mathematical Society , Providence, Rhode Island 2002, ISBN 0-8218-2968-8 ( MR1940576 ).
• Paul Kirchberger: About Chebyshev approaches . In: Mathematical Annals . tape 57 , 1903, pp. 509-540 ( MR1511222 ).
• Steven R. Lay: Convex Sets and Their Applications . John Wiley & Sons, New York / Chichester / Brisbane / Toronto / Singapore 1982, ISBN 0-471-09584-2 .
• Kurt Leichtweiß : Convex quantities (=  university text ). Springer Verlag, Berlin / Heidelberg / New York 1980, ISBN 3-540-09071-1 ( MR0586235 ).
• Jürg T. Marti : Convex Analysis (=  textbooks and monographs from the field of exact sciences, mathematical series . Volume 54 ). Birkhäuser Verlag , Basel, Stuttgart 1977, ISBN 3-7643-0839-7 ( MR0511737 ).
• Jan van Tiel: Convex Analysis . An Introductory Text. John Wiley & Sons, Chichester / New York / Brisbane / Toronto / Singapore 1984 ( MR0743904 ).
• RJ Webster: Another simple proof of Kirchberger's theorem . In: Journal of Mathematical Analysis and Applications . tape 92 , 1983, pp. 299-300 ( MR0694178 ).

References and comments

1. ↑ ^{a b} Alexander Barvinok: A Course in Convexity. 2002, p. 21 ff
2. ↑ Steven R. Lay: Convex Sets and Their Applications. 1982, p. 47 ff
3. ↑ Jürg T. Marti: Convex Analysis. 1977, p. 203 ff
4. ^ Jan van Tiel: Convex Analysis. 1984, p. 41 ff
5. ↑ Lay, op.cit., P. 56
6. ↑ Marti, op.cit., P. 205
7. ↑ van Tiel, op.cit., P. 44
8. ↑ Kurt Leichtweiß: Convex sets. 1980, p. 74 ff
9. ↑ See entry in the Mathematics Genealogy Project !