Unit sphere

Under the unit sphere is understood in mathematics, the sphere of radius one around the zero point of a vector space. Here, a generalized concept of distance is taken as a basis, so that depending on context the unit sphere must have no more resemblance to a conventional ball. This unit sphere is the boundary of the unit sphere, in the two-dimensional real vector space with the Euclidean norm is the unit circle.

General definition

It should be a normed vector space. Then we call the set of points whose distance from the origin is less than one, the open unit ball in:

According referred

The closed unit ball in and

The unit sphere in.

By translation and scaling can be converted any balls in the unit sphere in a room. Therefore, it is often sufficient to prove certain statements only for the unit sphere in order to infer the validity of any balls.

Unit sphere in finite-dimensional spaces

In the case of Euclidean space one defines the closed unit ball with respect to the Euclidean norm by

The sum norm (1- norm ) and the maximum norm another unit balls can be defined also with respect. The geometric shape of the unit sphere depends on the selected norm and the Euclidean norm is actually spherical.

Volume

The volume of an -dimensional Euclidean unit ball

Here, the gamma function, an analytic continuation of the ( shifted ) is on the faculty at the real numbers. For straight, the formula simplifies to.

It is worth noting in this context that the volume of the unit sphere increases as a function of the space dimension to first, and drops again - and even go for against 0:

Comments

• The unit sphere forms the edge of the unit sphere. According to the two-dimensional unit sphere is not the circle, but the circular disk.

• More generally, a unit sphere is defined in each metric space. It should be noted that there was not a point must be awarded as zero from the outset and we can speak of the unit ball of a metric space so limited. Furthermore Speaking of metrics that are non-standard induces the unit balls even further removed from the view. Especially true in a vector space with the discrete metric: , and.

• When viewed from environments, the unit sphere is referred to as 1- 1- sphere or ball.

Properties

• The closed unit ball is convex. ( The convexity follows from the triangle inequality. )

• It is point symmetric about the origin 0.

• Conversely, in a finite-dimensional vector space by any closed convex set, which is point symmetric about the origin and contains the origin in the interior, a standard defined that has this amount as a unit sphere: for (see Minkowski functional ).

• The closed unit ball is compact if it is finite-dimensional.

• The closed unit ball is weakly compact if and only if it is reflexive.

• The closed unit ball in the topological dual space of is always weak - * - compact ( Banach - Alaoglu ).

Applications in the Natural Sciences

In a variety of species, the unit sphere is used in the geosciences, particularly for computations on the globe. They are made with so-called spherical triangles and the formulas of Spherical Trigonometry, if an accuracy of about 0.1 % is sufficient, for example in geography and cartography, Globe calculations and navigation. The true distances are obtained from the ball arcs through multiplication by the earth's radius.

For higher accuracy - especially in geodesy - to be used in place of the globe Erdellipsoid. Using the method of Verebnung but also spherical triangle calculations are possible.

Geologists use for direction analysis of rock layers or fractures a unit sphere, which they call attitude sphere. In it the normal vectors of the respective planes are entered and then shown in equal-area azimuthal.

Also astronomical calculations are performed on the unit sphere to the observer since time immemorial. It corresponds to the freiäugigen sight of the heavens and the celestial sphere is called, on the spherical astronomy has defined its own coordinate systems for angle measurements and star positions. Whether the ball with radius 1 or ∞ is assumed irrelevant in this regard.

Swell

• Dirk Werner: Functional Analysis. 6, corrected edition, Springer -Verlag, Berlin 2007, ISBN 978-3-540-72533-6

• Functional Analysis