Kakutani-Yamabe-Yujobô theorem

The theorem of Kakutani-Yamabe-Yujobô is a mathematical theorem about the topology of the unit sphere in Euclidean space , which goes back to the mathematicians Shizuo Kakutani , Hidehiko Yamabe and Zuiman Yujobô and which is related to the theorem of Borsuk-Ulam . Kakutani showed the sentence in its original version for the unit sphere in three-dimensional Euclidean space and was thus able to confirm an open conjecture by Hans Rademacher (1892–1969) about the inclusion of compact convex bodies by cubes . The theorem was later extended to the multidimensional case by Yamabe and Yujobô .

The Kakutani-Yamabe-Yujobô theorem is also closely related to a result by Freeman Dyson and Chung-Tao Yang . As Yang showed in 1954, all these theorems can be proven using uniform methods of homology theory.

Formulation of the sentence

The sentence can be formulated as follows:

Let any dimension number and a continuous function be given , which assigns a real number to every point of the sphere . ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle f\colon \,S^{N-1}\to \mathbb {R} }$ ${\displaystyle P}$ ${\displaystyle (N-1)}$ ${\displaystyle S^{N-1}\subset \mathbb {R} ^{N}}$ ${\displaystyle f(P)}$

Then there are always points with the same value such that the associated radius vectors are at right angles to one another in pairs . ${\displaystyle N}$ ${\displaystyle P_{1},\dots ,P_{N}\in S^{N-1}}$ ${\displaystyle f}$ ${\displaystyle f(P_{1})=\dots =f(P_{N})}$ ${\displaystyle {\overrightarrow {O{P_{1}}}},\dots ,{\overrightarrow {O{P_{N}}}}}$

The proof of the theorem can be carried out within the framework of the homology theory according to Eduard Čech and Paul A. Smith .

In this case , this result can also be described as follows: ${\displaystyle N=3}$

If a continuous real function is given on the spherical surface of the unit sphere in three-dimensional Euclidean space, the spherical surface always contains an equilateral spherical triangle of side length , whose corners all have the same function value. ${\displaystyle \pi /2}$

Conclusion: Hans Rademacher's assumption

The correctness of the "assumption of Hans Rademacher" results from the following corollary to the "sentence of Kakutani-Yamabe-Yujobô":

A non-empty compact set in the n-dimensional Euclidean space is always in an N-dimensional cube so contained that any -dimensional side of at least one contact point with in common. ${\displaystyle {\mathcal {K}}}$ ${\displaystyle N}$ ${\displaystyle {\mathcal {W}}}$ ${\displaystyle (N-1)}$ ${\displaystyle {\mathcal {W}}}$ ${\displaystyle {\mathcal {K}}}$

From this follows the theorem assumed by Rademacher for the number of dimensions : ${\displaystyle N=3}$

Each non-empty compact (convex) portion of the three-dimensional Euclidean space can be of a cube in such a way include that they with each face of the cube has a common contact point.

Derivation of the conclusion (evidence sketch)

One considers as fixed and then for each point all hyperplanes that run orthogonally to the associated radius vector . ${\displaystyle {\mathcal {K}}\neq \emptyset }$ ${\displaystyle P\in S^{N-1}}$ ${\displaystyle {\overrightarrow {O{P}}}}$

Among them are two parallel hyperplanes and , which touch each other and thereby form the edge of a closed space segment , which encompasses such that the Euclidean distance between the two hyperplanes is the smallest possible (among all possible distances between two hyperplanes constructed in this way). ${\displaystyle {\mathcal {H}}_{1}}$ ${\displaystyle {\mathcal {H}}_{2}}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle {\mathcal {K}}}$

This distance is a non-negative real value and is to be understood as the width of the space segment bounded by and ${\displaystyle {\mathcal {H}}_{1}}$ ${\displaystyle {\mathcal {H}}_{2}}$ , thus as the width of in the direction ${\displaystyle {\mathcal {K}}}$ ${\displaystyle {\overrightarrow {O{P}}}}$ . If this value is designated with , then there is a continuous real function on the . ${\displaystyle d(P)}$ ${\displaystyle P\mapsto d(P)}$ ${\displaystyle S^{N-1}}$

For this continuous function one now applies the "Theorem of Kakutani-Yamabe-Yujobô". In the given situation it means that there are closed space segments of identical width, which encompass all and whose associated radius vectors are perpendicular to each other in pairs. However, this means that the intersection of these space segments forms a -dimensional cube. Since all of these include space segments , this is the cube we are looking for. ${\displaystyle N}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle {\mathcal {K}}}$

Related result: Dyson-Yang Theorem

The "Theorem of Dyson-Yang" makes the following statement:

For a continuous real function on the sphere ( ) there are always diameters which are pairwise at right angles to each other and whose end points all have the same function value. ${\displaystyle n}$ ${\displaystyle S^{n}\subset {\mathbb {R} }^{n+1}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle n}$ ${\displaystyle 2n}$

If you put the number of dimensions here , this leads to the original "Dyson theorem": ${\displaystyle n=2}$

If a continuous real function is given on a 2-sphere in three-dimensional Euclidean space, the 2-sphere always contains four points which form a square on a great circle of this 2-sphere and which all assume the same function value.

literature

Original work

• FJ Dyson : Continuous functions defined on spheres . In: Ann. Math . tape 54 , 1942, pp. 534-536 , JSTOR : 1969487 ( MR0044620 ).
• Shizuo Kakutani: A proof that there exists a circumscribing cube around any bounded closed convex set in R ³ . In: Ann. Math . tape 43 , 1942, pp. 739-741 , JSTOR : 1968964 ( MR0007267 ).
• Hidehiko Yamabe and Zuiman Yujobô: On the continuous function defined on a sphere . In: Osaka Math J. . tape 2 , 1950, p. 19–22 ( projecteuclid.org [PDF; 465 kB ]). MR0037006
• Chung-Tao Yang: On theorems of Borsuk-Ulam Kakutani-Yamabe Yujobô and Dyson, I . In: Ann. Math . tape 60 , 1954, pp. 262-282 , JSTOR : 1969632 ( MR0065910 ).

Monographs

• Max K. Agoston: Algebraic Topology: A First Course (=  Monographs and Textbooks in Pure and Applied Mathematics . Volume 32 ). Marcel Dekker, New York [a. a.] 1976, ISBN 0-8247-6351-3 ( MR0445485 ).
• Arlo W. Schurle: Topics in Topology (=  Monographs and Textbooks in Pure and Applied Mathematics . Volume 32 ). North Holland, New York-Oxford 1979, ISBN 0-444-00285-5 ( MR0542116 ).

Individual evidence

1. ↑ Yang: Ann. Math . tape 60 , p. 262 ff .
2. ↑ ^{a b} Agoston: p. 245.
3. ↑ Yang: Ann. Math . tape 60 , p. 262 .
4. ↑ Yang: Ann. Math . tape 60 , p. 263 ff .
5. ↑ See Schurle: p. 164 ff .; according to Schurle this is the "sentence of Kakutani"
6. ^ Agoston: p. 246.
7. ↑ In the English-speaking world, this result is also sometimes given as “Kakutani's sentence”, as is the case here .
8. ↑ The requirement of convexity turns out to be unnecessary.
9. ↑ In the case of the Euclidean plane, the hyperplanes are straight lines and a closed space segment of the kind described is an infinite flat strip.
10. ↑ Dyson: Ann. Math . tape 54 , p. 534 ff .
11. ↑ Yang: Ann. Math . tape 60 , p. 282 .
12. ↑ A diameter is therefore the connecting distance between two opposing spherical points. The two points are also referred to as antipodes . For a circular line in the Euclidean plane, a diameter is therefore a chord of maximum length.
13. ↑ Dyson: Ann. Math . tape 54 , p. 534 ff .