In
mathematics, the
Borsuk–Ulam theorem states that every
continuous function from an
n-sphere into
Euclidean n-space maps some pair of
antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.
Formally: if {\displaystyle f:S^{n}\to \mathbb {R} ^{n}}
[/URL] is continuous then there exists an {\displaystyle x\in S^{n}}
[/URL] such that: {\displaystyle f(-x)=f(x)}
[/URL].
The case {\displaystyle n=1}
[/URL] can be illustrated by saying that there always exist a pair of opposite points on the
Earth's equator with the same temperature. The same is true for any circle. This assumes the temperature varies continuously in space.
The case {\displaystyle n=2}
[/URL] 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, assuming that both parameters vary continuously in space.
The Borsuk–Ulam theorem has several equivalent statements in terms of
odd functions. Recall that {\displaystyle S^{n}}
[/URL] is the
n-sphere and {\displaystyle B^{n}}
[/URL] is the
n-ball:
- If {\displaystyle g:S^{n}\to \mathbb {R} ^{n}}[/URL] is a continuous odd function, then there exists an {\displaystyle x\in S^{n}}[/URL] such that: {\displaystyle g(x)=0}[/URL].
- If {\displaystyle g:B^{n}\to \mathbb {R} ^{n}}[/URL] is a continuous function which is odd on {\displaystyle S^{n-1}}[/URL] (the boundary of {\displaystyle B^{n}}[/URL]), then there exists an {\displaystyle x\in B^{n}}[/URL] such that: {\displaystyle g(x)=0}[/URL].
