Straszewicz's theorem

The set of Straszewicz ( English Straszewicz's theorem ) is a theorem of the mathematical area of convex geometry and as such located between the fields of geometry and analysis . It goes back to a scientific work of the mathematician Stefan Straszewicz from the year 1935. The Straszewicz'sche sentence is related to the Kerin-Milman theorem and deals with the question of what relationship in Euclidean space the exposed points and the extreme points of certain point sets another stand. As the theorem shows, for a large class of point sets the exposed points form a dense subset within the extremal points.

Formulation of the sentence

The sentence can be summarized as follows:

The following always applies to a closed and convex subset : ${\displaystyle T\subseteq \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$

(i) Every extreme point of is the contact point of the set of exposed points of : ${\displaystyle T}$ ${\displaystyle T}$

${\displaystyle \operatorname {ext} {T}\subseteq {\overline {\operatorname {exp} {T}}}}$.

(ii) If this is a convex compact , then it even holds: ${\displaystyle T}$

${\displaystyle T={\overline {\operatorname {conv} \left(\operatorname {exp} {T}\right)}}}$.

Analog for standardized spaces

The American mathematician Victor Klee has in 1958 presented a set of the Straszewicz analogous result for the more general case where a normalized - vector space exists. This result is called the Klee – Straszewicz theorem and can be stated as follows: ${\displaystyle \mathbb {R} }$

In a normalized - vector space holds for every compact and convex subset contained therein ${\displaystyle \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle T\subseteq X}$

${\displaystyle \operatorname {ext} {T}\subseteq {\overline {\operatorname {exp} {T}}}}$

and

${\displaystyle T={\overline {\operatorname {conv} \left(\operatorname {exp} {T}\right)}}}$.

Explanations and Notes

• An exposed point of is a point to which a - supporting hyperplane exists such that holds. The set of exposed points is denoted by. ${\displaystyle T}$ ${\displaystyle p\in T}$ ${\displaystyle T}$ ${\displaystyle H\subset X}$ ${\displaystyle H\cap T=\{p\}}$ ${\displaystyle T}$ ${\displaystyle \operatorname {exp} {T}}$
• For a convex subset of , each of its exposed points is always also an extremal point and each of its extremal points is always also one of its edge points . So it applies in this case . ${\displaystyle X}$ ${\displaystyle \operatorname {exp} {T}\subseteq \operatorname {ext} {T}\subseteq \partial {T}}$
• The set of Straszewicz is in the monograph by Kurt Leichtweiß as a display set of Straszewicz denotes (wherein Leichtweiß refers only to the above amount equation).

literature

• Arne Brøndsted : An Introduction to Convex Polytopes (=  Graduate Texts in Mathematics . Volume 90 ). Springer-Verlag , New York, Heidelberg, Berlin 1983, ISBN 0-387-90722-X ( MR0683612 ).
• Branko Grünbaum : Convex Polytopes (=  Graduate Texts in Mathematics . Volume 221 ). Springer-Verlag, New York, Berlin, Heidelberg 2003, ISBN 0-387-00424-6 ( MR1976856 ).
• Victor L. Klee, Jr .: Extremal structure of convex sets. II . In: Mathematical Journal . tape 69 , 1958, pp. 90-104 , doi : 10.1007 / BF01187394 ( MR0092113 ).
• 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 ).
• S. Straszewicz: Set of points closed over exposed points . In: Fundamenta Mathematicae . tape 24 , 1935, pp. 139-143 .

Individual evidence

1. ↑ Kurt Leichtweiß: Convex sets. 1980, pp. 35-45
2. ↑ Jürg T. Marti: Convex Analysis. 1977, pp. 94-97
3. ↑ ^{a b} Arne Brøndsted: An Introduction to Convex Polytopes. 1983, p. 37
4. ^ Branko Grünbaum: Convex Polytopes. 2003, p. 19
5. ↑ Leichtweiß, op. Cit. , Pp. 42-43
6. ↑ Marti, op.cit., P. 94, p. 97
7. ↑ Victor L. Klee, Jr .: Extremal structure of convex sets. II. Math. Z. 69, p. 91
8. ↑ Marti, op. Cit., Pp. 125-130
9. ↑ Leichtweiß, op. Cit. , P. 41
10. ↑ Marti, op.cit., P. 34, p. 90
11. ↑ Marti, op.cit., P. 34, p. 91
12. ↑ Leichtweiß, op. Cit. , P. 42