Borsuk–Ulam theorem

The Borsuk -Ulam theorem states that any continuous function of a sphere in - dimensional Euclidean space is a pair of antipodal points mapping to the same point. ( Two points of a sphere are called antipodal if they are in exactly opposite directions from the center. )

The case is often characterized explains that at any time a pair of antipodal points on the earth's surface at the same temperature and the same air pressure exist. This assumes that the temperature and pressure are continuous functions.

The Borsuk -Ulam theorem was conjectured by Stanisław Ulam and Karol Borsuk proved 1933. It is possible to derive the brouwer between fixed point theorem of the Borsuk -Ulam theorem in an elementary way.

Borsukscher antipodes set

A stronger statement is the Borsuk, which is also known as Borsukscher antipodes sentence. We call a function antipodes preserving if it is odd.

Statement

Is a symmetric, open and bounded subset of, containing the origin, and steadily and antipodal -preserving, ie, for all, as well. Then the Brouwer mapping degree is an odd number.

Further generalizations

• The statement can be generalized also to infinite-dimensional normed spaces. It should be a balanced, open and bounded subset of the normed space, a compact figure, and

Application

In elementary geometry can prove the following interesting fact with the statement of Borsuk - Ulam: (Also known as Bisektionstheorem )

" Given two arbitrary polygons in the plane. Then there is a straight line such that they bisect the area of the two polygons simultaneously. (ie taken not only in the sum but even both to himself) "