A new and constructive proof of the Borsuk-Ulam theorem

Abstract

The Borsuk-Ulam Theorem [1] states that if f is a continuous function from the n-sphere to n-space ( f : S n → R n ) (f:{S^n} \to {{\mathbf {R}}^n}) then the equation f ( x ) = f ( − x ) 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.