Continuous Functions Defined on Spheres¶
Authors: F. J. Dyson
Published: 1951 (Journal Paper)
Source: The Annals of Mathematics
Algorithm: Dyson's Sphere Theorem
DOI: 10.2307/1969487
Summary¶
Dyson proves a Borsuk-Ulam-style theorem for continuous real-valued functions on the 2-sphere: there are four points on the sphere, forming the vertices of a square centered at the sphere's center, at which the function takes the same value. Equivalently, the theorem finds two perpendicular diameters of the sphere whose four endpoints share a common function value. The result sits in the line of Kakutani and Yamabe-Yujobo sphere theorems used in geometric existence problems, and Dyson's proof adapts their topological method through reflection and connectedness arguments rather than giving a constructive way to locate the square. The paper is also historically useful because it identifies the natural higher-dimensional conjecture, equal values on the endpoints of several mutually perpendicular diameters, while leaving that extension to later work by Yang.
Abstract¶
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Tags¶
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Algebraic topology
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Borsuk-Ulam theorem
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Sphere theorems
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Continuous functions
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Coincidence theorems
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Inscribed square
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Topological combinatorics