Borsuk fixed-point theorem

Let $B$ be an open bounded symmetric subset of $\mathbf{R}^n$ containing the origin. Here, symmetric means that if $x \in B$, then $-x \in B$ also. Let $\phi : \partial B \rightarrow \mathbf{R}^m$ be a continuous mapping and let $m < n$. Then there is an $x \in \partial B$ such that $\phi(x) = \phi(-x)$.

The original version (K. Borsuk, 1933) was for $B = D^n$, the $n$-dimensional ball in $\mathbf{R}^n$, $n = m$. The result is also known as one of the Borsuk antipodal theorems (see Antipodes) or as the Borsuk–Ulam theorem.

The central lemma for the Borsuk–Ulam theorem is that an odd mapping $f : S^n \rightarrow S^n$ has odd degree (see Degree of a mapping). A mapping $f : S^n \rightarrow S^n$ is called odd if $f(-x) = - f(x)$. Many people call this odd-degree result itself the Borsuk–Ulam theorem. For a generalization, the so-called Borsuk odd mapping theorem, see [a1], p. 42.

References

[a1]	N.G. Lloyd, "Degree theory" , Cambridge Univ. Press (1978)
[a2]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. 266