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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.


References [Enhancements On Off] (What's this?)

  • [1] Karol Borsuk, Drei Sätze über die n-dimensional euklidische Sphäre, Fund. Math. 20 (1933), 177-190.
  • [2] B. Curtis Eaves, A short course in solving equations with PL homotopies, Nonlinear programming (Proc. Sympos., New York, 1975), SIAM-AMS Proc., Vol. IX, Amer. Math. Soc., Providence, R.I., 1976, pp. 73-143. MR 0401155 (53:4984)
  • [3] B. Curtis Eaves and H. Scarf, The solution of systems of piecewise linear equations, Mathematics of Operations Research 1 (1976), 1-27. MR 0445792 (56:4126)
  • [4] E. Pannwitz, Eine freie Abbildung der n-dimensional Sphäre in die Ebene, Math. Nachr. 7 (1952), 183-185. MR 0048021 (13:965i)
  • [5] Colin P. Rourke and Brian J. Sanderson, Introduction to piecewise-linear topology, Springer-Verlag, Berlin and New York, 1972. MR 665919 (83g:57009)

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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

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