# Borsuk-Ulam theorem

The**Borsuk-Ulam theorem**states that any continuous function from an

*n*-sphere into Euclidean

*n*-space maps some pair of antipodal points to the same point. (Two points on a sphere are called antipodal if they sit on directly opposite sides of the sphere's center.)

The case *n* = 2 is often illustrated by saying that at any moment there is always a pair of antipodal points on the Earth's surface with equal temperature and equal barometric pressure. This assumes that temperature and barometric pressure vary continuously.

The Borsuk-Ulam theorem was first conjectured by Stanislaw Ulam. It was proved by Karol Borsuk in 1933.

## References

- K. Borsuk, "Drei Sätze über die n-dimensionale euklidische Sphäre",
*Fund. Math.*,**20**(1933), 177-190. - Jiří MatouÚek,
*"Using the Borsuk-Ulam theorem"*, Springer Verlag, Berlin, 2003. ISBN 3-540-00362-2. - L. Lyusternik and S. Shnirel'man, "Topological Methods in Variational Problems".
*Issledowatelskii Institut Matematiki i Mechaniki pri O. M. G. U.*, Moscow, 1930.