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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A new and constructive proof of the Borsuk-Ulam theorem


Authors: Mark D. Meyerson and Alden H. Wright
Journal: Proc. Amer. Math. Soc. 73 (1979), 134-136
MSC: Primary 55M20
DOI: https://doi.org/10.1090/S0002-9939-1979-0512075-9
MathSciNet review: 512075
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Abstract: The Borsuk-Ulam Theorem [1] states that if f is a continuous function from the n-sphere to n-space $ (f:{S^n} \to {{\mathbf{R}}^n})$ then the equation $ f(x) = f( - x)$ has a solution. It is usually proved by contradiction using rather advanced techniques. We give a new proof which uses only elementary techniques and which finds a solution to the equation. If f is piecewise linear our proof is constructive in every sense; it is even easily implemented on a computer.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0512075-9
Keywords: Borsuk-Ulam Theorem, Ham Sandwich Theorem, Invariance of Domain, fixed point, antipodal map, piecewise linear, simplicial methods, complementary pivot theory
Article copyright: © Copyright 1979 American Mathematical Society