Motzkin theorem

The set of Motzkin is a mathematical theorem that a work of mathematician Theodore Samuel Motzkin back from the year 1935th It deals with the question of the characterization of convex subsets of Euclidean space and is located in the transition field between analysis , geometry and the theory of topological vector spaces .

Formulation of the sentence

Following the monograph by Jürg T. Marti , the sentence can be formulated as follows:

Im a Motzkin set is always convex . ${\displaystyle \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$

generalization

In 1951 Frederick Arthur Ficken and Victor LaRue Klee obtained the following characterization theorem for convex sets in real Hilbert spaces as a generalization of Motzkin's theorem :

Every bounded compact Motzkin set in a real Hilbert space is convex.

Explanations and Notes

• If there is a metric space with the associated distance function , a non-empty closed subset is called a Motzkin set if there is exactly one point in space for each point in space , which is closest to the point in space according to the distance function . Some authors also call such a set a Chebyshev set . ${\displaystyle X}$ ${\displaystyle d}$ ${\displaystyle T\subseteq X}$ ${\displaystyle x_{0}\in X}$ ${\displaystyle x_{1}\in T}$ ${\displaystyle d}$ ${\displaystyle x_{0}}$
• In a metric space with the distance function , a non-empty closed subset is therefore a Motzkin set if and only if there is exactly one with for each . If there is even a normalized vector space with as norm and the given distance function by, then here a non-empty closed subset is a Motzkin set if and only if there is exactly one with for each . ${\displaystyle X}$ ${\displaystyle d}$ ${\displaystyle T\subseteq X}$ ${\displaystyle x_{0}\in X}$ ${\displaystyle x_{1}\in T}$ ${\displaystyle d(x_{1},x_{0})=\inf _{y\in T}{d(y,x_{0})}}$ ${\displaystyle X}$ ${\displaystyle \|\cdot \|}$ ${\displaystyle (x,y)\mapsto d(x,y)=\|x-y\|}$ ${\displaystyle T\subseteq X}$ ${\displaystyle x_{0}\in X}$ ${\displaystyle x_{1}\in T}$ ${\displaystyle \|x_{1}-x_{0}\|=\inf _{y\in T}{\|y-x_{0}\|}}$
• Euclidean space is always considered to be provided with the standard scalar product and the geometrical and metric structure given with it. ${\displaystyle \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$
• In a normed vector space is called - according Marti - a subset limited compact if for every natural number which - subset one there compact subset is. ${\displaystyle X}$ ${\displaystyle K\subseteq X}$ ${\displaystyle n}$ ${\displaystyle X}$ ${\displaystyle \{x\in K\colon \|x\|\leq n\}}$
• In a strictly convex normalized space , every non-empty compact convex subset is a Motzkin set.
• In a strictly convex reflexive Banach space - and consequently also in every Hilbert space - every non-empty closed convex subset is a Motzkin set.
• Motzkin's theorem can be derived from Blaschke's selection theorem .
• In his textbook Convex Quantities , Kurt Leichtweiß evaluates Motzkin's theorem - although he does not expressly represent it under this name - as a remarkable characterization of the convexity in closed subsets of Euclidean space, originating from TS Motzkin .

literature

• Victor Klee: Convexity of Chevyshev sets . In: Mathematical Annals . tape 142 , 1961, pp. 292-304 ( MR0121633 ).
• Steven R. Lay: Convex Sets and Their Applications (=  Pure and Applied Mathematics ). John Wiley & Sons , New York, Chichester, Brisbane, Toronto, Singapore 1982, ISBN 0-471-09584-2 ( MR0655598 ).
• 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 ).
• TS Motzkin: Sur quelques propriétés caractéristiques des ensembles convexes . In: Atti. Real Accad. Naz. Lincei, Rend. Cl. Sci. F sharp. Mat. Nat. (Rome), Series VI . tape 21 , 1935, pp. 562-567 .
• Frederick A. Valentine : Convex Sets . Translation from English by E. Heil (=  BI university paperbacks . Volume 402 / 402a). Bibliographisches Institut , Mannheim 1968 ( MR0226495 ).

Individual evidence

1. ↑ Jürg T. Marti: Convex Analysis. 1977, pp. 153-158
2. ↑ Steven R. Lay: Convex Sets and Their Applications. 1982, p. 53
3. ↑ Frederick A. Valentine: Convex sets. 1968, pp. 103-107, p. 185
4. ↑ Kurt Leichtweiß: Convex sets. 1980, pp. 95-97
5. ↑ Marti, op.cit., P. 158
6. ↑ FA Ficken (August 13, 1910-20 December 1978) was an American mathematician (see link ) and editor of the American Mathematical Monthly for the period 1962-1966.
7. ↑ Marti, op.cit., P. 156
8. ↑ This characterization sentence and its derivation are, as Victor Klee expressly states in his publication from 1961, essentially to be attributed to FA Ficken. In Marti's monograph (see p. 156 and p. 271) the sentence is referred to as the Ficken-Klee sentence .
9. ↑ Valentine, op.cit., P. 185
10. ↑ Marti, op.cit., P. 153
11. ↑ Lay, op.cit., P. 112
12. ↑ ^{a b c} Marti, op.cit., P. 154
13. ↑ Marti, op.cit., P. 158
14. ↑ Leichtweiß, op.cit., P. 95