Generalization to metric spaces
More generally, in a metric space In mathematics, a metric space is a set where a notion of distance between elements of the set is defined (E,d), the sphere of center x and radius r > 0 is the set of points y such that d(x,y) = r.
If the center is a distinguished point considered as origin of E, as in a normed In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector. A seminorm , on the other hand, is allowed to assign zero length to some non-zero vectors space, it is not mentioned in the definition and notation. The same applies for the radius if it is taken equal to one, as in the case of a unit sphere.
In contrast to a ball In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general, a sphere may be an empty set, even for a large radius. For example, in Zn with Euclidean metric In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. By using this formula as distance, Euclidean space becomes a metric space . The associated norm is called the Euclidean norm, a sphere of radius r is nonempty only if r2 can be written as sum of n squares of integers.
<<Table of Contents A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius of the sphere. The | Next>> | Show All>>
