Borsuk-Ulam's Theorem
The Borsuk-Ulam theorem states that every continuous function of a - sphere in the -dimensional Euclidean space maps a pair of antipodal points to the same point. (Two points of a sphere are called antipodal if they lie in exactly opposite directions from the center point.)
The case is often explained by the fact that at any point in time a pair of antipodal points exist on the earth's surface with the same temperatures and the same air pressure. This assumes that temperature and air pressure are continuous functions.
Borsuk-Ulam's theorem was conjectured by Stanisław Ulam and proved by Karol Borsuk in 1933 . It is possible to derive Brouwer's fixed point theorem in an elementary way from the Borsuk-Ulam theorem . There are various generalizations of the sentence, so that one speaks of sentences of the Borsuk-Ulam type.
statement
There are several equivalent formulations of the sentence:
- Let be a continuous antipodal map, then is . Antipodal means that applies to everyone .
- Let be a continuous antipodal map. Then there is one with
- Be a continuous map. Then there is a fixed point with . This is the formulation in the introduction.
- If the n-sphere is covered by (n + 1) open or closed subsets of the n-sphere, at least one of the two contains an antipodal pair of points.
Borsuk antipode theorem
A stronger statement is Borsuk's theorem, also known as Borsuk's antipodal theorem. A function is called antipode-preserving if it is odd .
statement
Is a symmetrical, open and bounded subset of des , which contains the zero point, and is continuous and antipode-preserving, i.e. for all , as well as . Then Brouwer's degree of mapping is an odd number.
Further generalizations
- Instead of demanding that it be antipodal, it is enough
- and to demand. Functions that fulfill this are homotopic to an antipode-preserving function, which is sufficient for the proof of Borsuk's theorem. In particular, there is no continuous continuation from on with . Because if Brouwer's degree of mapping is not equal to zero, then the equation has at least one solution .
- The statement can also be generalized to infinitely dimensional normalized spaces . It should be a symmetrical, open and bounded subset of the normalized space , wherein a compact image is, and
- Then the Leray shudder degree is an odd number.
application
In elementary geometry , the statement of Borsuk-Ulam can prove the following interesting fact (also known as Stone-Tukey's theorem or Ham sandwich theorem ):
"Given any two polygons in the plane. Then a straight line exists such that it halves the area of both polygons at the same time . (ie not only in total, but even both separately) "
- proof
- Let be and denote the given polygons with. Consider this in the shifted - -plane that we consider in standard Euclidean space. Let then be the position vector of a point on the unit sphere and denote with the normal plane to through the zero point. For the intersection of with defines a straight line . This is precisely illustrations can be explained by virtue of the continuous mapping: . Apparently these images have the property . If the measure of a content denotes, another continuous mapping of can be explained with the definition . Borsuk-Ulam then provides for a point the existence of . According to the construction of , this point applies to both . The straight line we are looking for is thus from the assertion.
The sentence is also used in topological combinatorics . There the theorem is closely related to Tucker's lemma and is equivalent to it. Sometimes the sentence of Borsuk-Ulam is used there in a variant or generalization by Albrecht Dold .
literature
- Karol Borsuk: Three sentences about the -dimensional Euclidean sphere . Fundamenta Mathematicae 20 (1933), 177-190, online
- Klaus Deimling: Nonlinear Functional Analysis . 1st edition. Springer-Verlag, Berlin / Heidelberg 1985, ISBN 3-540-13928-1 .
- Wolfgang Gromes: A simple proof of Borsuk's theorem . Mathematische Zeitschrift 178 (1981), 399-400, online .
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Lasar Ljusternik and Lew Schnirelmann : Topological Methods in Variational Problems . Issledowatelskii Institut Matematiki i Mechaniki pri OMGU, Moscow 1930 (Russian).
French translation by J. Kravtchenko: Méthodes topologiques dans lesproblemèmes variationnels. 1ère partie. Espaces à un nombre fini de dimensions . Hermann & Cie., Paris 1934. - Jiří Matoušek: Using the Borsuk-Ulam theorem . Springer-Verlag, Berlin 2003, ISBN 3-540-00362-2 .
Web links
- Allen Hatcher: Algebraic Topology (Eng.)
- Derivation of Brouwer's Fixed Point Theorem using the Borsuk-Ulam theorem (PDF file; 125 kB)
- Video: Borsuk antipodal theorem . Institute for Scientific Film (IWF) 1982, made available by the Technical Information Library (TIB), doi : 10.3203 / IWF / C-1433 .