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 are in exactly opposite directions from 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 temperatures and equal barometric pressures. 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 Matoušek, "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.