Krasnoselski's theorem

The set of Krasnosel'skii ( English Krasnosselsky's theorem or Krasnoselsky's theorem or Krasnosel'skii's theorem ) is one of the classic tenets of the mathematical part of the area of the convex geometry and, as such, located in the transition field between geometry and Analysis . It goes back to a scientific work by the Soviet mathematician Mark Alexandrowitsch Krasnoselski from 1946. The sentence deals with the question under which conditions certain subsets of Euclidean space are star-shaped . It is related to (and even a consequence of) Helly's theorem .

Formulation of the sentence

The sentence can be summarized as follows:

A natural number and a compact subset consisting of an infinite number of spatial points are given . Here there is to each of points in space existing subset an associated point in space such that each of from visible (see Fig. U.) Is. ${\displaystyle n}$ ${\displaystyle X\subset \mathbb {R} ^{n}}$ ${\displaystyle n+1}$ ${\displaystyle T\subset X}$ ${\displaystyle p_{T}\in X}$ ${\displaystyle t\in T}$ ${\displaystyle p_{T}}$

Then:

${\displaystyle X}$ is star-shaped.

Addition : The assertion of the theorem is still valid even if the above visibility condition is weakened and it is only required for each subset consisting of regular (see below) spatial points . ${\displaystyle n+1}$ ${\displaystyle T\subset X}$

Explanations

• For two points is of of (in ) visible - and vice versa! - if their connecting line is a subset of , i.e. if the relation holds for their convex hull . ${\displaystyle p,q\in X}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle \operatorname {conv} \{p,q\}\subseteq X}$
• An ordinary point of is an edge point that is also a support point of . It is a support point from a point in space to which a linear functional exists, which is not the zero mapping and thereby fulfills the relationship . ${\displaystyle X}$ ${\displaystyle p\in \partial {X}}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle p\in \mathbb {R} ^{n}}$ ${\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }$ ${\displaystyle \sup _{x\in X}{f(x)}=f(p)}$

literature

• MA Krasnoselski: Sur un critère pour qu'un domaine soit étoilé . In: Matematitscheskii sbornik (NS) . tape 19 , 1946, pp. 309-310 ( MR0020248 ).
• 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 ).
• Frederick A. Valentine : Convex Sets (=  McGraw-Hill Series in Higher Mathematics ). McGraw-Hill Book Co. , New York, Toronto, London 1964 ( MR0170264 ).

Individual evidence

1. ^ ^{A b} Steven R. Lay: Convex Sets and Their Applications. 1982, p. 53
2. ↑ Jürg T. Marti: Convex Analysis. 1977, p. 203 ff
3. ↑ Kurt Leichtweiß: Convex sets. 1980, p. 76 ff
4. ↑ Frederick A. Valentine: Convex Sets. 1964, p. 82 ff
5. ↑ Leichtweiß, op. Cit., Pp. 76-77
6. ↑ Valentine, op.cit., P. 84
7. ↑ Marti, op.cit., P. 212
8. ↑ Marti, op.cit., P. 66, p. 211