A new and constructive proof of the Borsuk-Ulam theorem

Meyerson, Mark D.; Wright, Alden H.

doi:10.1090/S0002-9939-1979-0512075-9

A new and constructive proof of the Borsuk-Ulam theorem
HTML articles powered by AMS MathViewer

by Mark D. Meyerson and Alden H. Wright PDF

Proc. Amer. Math. Soc. 73 (1979), 134-136 Request permission

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

Drei Sätze über die n-dimensional euklidische Sphäre

20

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
B. Curtis Eaves and Herbert Scarf, The solution of systems of piecewise linear equations, Math. Oper. Res. 1 (1976), no. 1, 1–27. MR 445792, DOI 10.1287/moor.1.1.1
Erika Pannwitz, Eine freie Abbildung der $n$-dimensionalen Sphäre in die Ebene, Math. Nachr. 7 (1952), 183–185 (German). MR 48021, DOI 10.1002/mana.19520070306
Colin Patrick Rourke and Brian Joseph Sanderson, Introduction to piecewise-linear topology, Springer Study Edition, Springer-Verlag, Berlin-New York, 1982. Reprint. MR 665919