Ball (mathematics)

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A ball is the inside of a sphere

In mathematics, a ball is the space inside a sphere. It may be a closed ball (including the boundary points of the sphere) or an open ball (excluding them).



Balls in general metric spaces[edit]


B_r(p) \triangleq \{ x \in M \mid d(x,p) < r \}.

The closed (metric) ball, which may be denoted by Br[p] or B[pr], is defined by

B_r[p] \triangleq \{ x \in M \mid d(x,p) \le r \}.


The closure of the open ball Br(p) is usually denoted \overline{ B_r(p) }. While it is always the case that B_r(p) \subseteq \overline{ B_r(p) } and \overline{ B_r(p) } \subseteq B_r[p], it is not always the case that \overline{ B_r(p) } = B_r[p]. For example, in a metric space X with the discrete metric, one has \overline{B_1(p)} = \{p\} and B_1[p] = X, for any p \in X.

A (open or closed) unit ball is a ball of radius 1.


The open balls of a metric space are a basis for a topological space, whose open sets are all possible unions of open balls. This space is called the topology induced by the metric d.

Balls in normed vector spaces[edit]

Any normed vector space V with norm |·| is also a metric space, with the metric d(xy) = |x − y|. In such spaces, every ball Br(p) is a copy of the unit ball B1(0), scaled by r and translated by p.

Euclidean norm[edit]

In particular, if V is n-dimensional Euclidean space with the ordinary (Euclidean) metric, every ball is the interior of a hypersphere (a hyperball). That is a bounded interval when n = 1, the interior of a circle (a disk) when n = 2, and the interior of a sphere when n = 3.


In Cartesian space \R^n with the p-norm Lp, an open ball is the set

B(r) = \left\{ x \in \R^n \,:\, \sum_{i=1}^n \left|x_i\right|^p < r^p \right\}.

For n=2, in particular, the balls of L1 (often called the taxicab or Manhattan metric) are squares with the diagonals parallel to the coordinate axes; those of L (the Chebyshev metric) are squares with the sides parallel to the coordinate axes. For other values of p, the balls are the interiors of Lamé curves (hypoellipses or hyperellipses).

For n = 3, the balls of L1 are octahedra with axis-aligned body diagonals, those of L are cubes with axis-aligned edges, and those of Lp with p > 2 are superellipsoids.

General convex norm[edit]


Topological balls[edit]


Any open topological n-ball is homeomorphic to the Cartesian space Rn and to the open unit n-cube (0,1)^n \subseteq \R^n. Any closed topological n-ball is homeomorphic to the closed n-cube [0, 1]n.



See also[edit]


  • D. J. Smith and M. K. Vamanamurthy, "How small is a unit ball?", Mathematics Magazine, 62 (1989) 101–107.
  • "Robin conditions on the Euclidean ball", J. S. Dowker [1]
  • "Isometries of the space of convex bodies contained in a Euclidean ball", Peter M. Gruber[2]