Ball (mathematics)

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In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere.^[1] It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).

These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment.

In other contexts, such as in Euclidean geometry and informal use, sphere is sometimes used to mean ball. In the field of topology the closed ${\displaystyle n}$-dimensional ball is often denoted as ${\displaystyle B^{n}}$ or ${\displaystyle D^{n}}$ while the open ${\displaystyle n}$-dimensional ball is ${\displaystyle \operatorname {int} B^{n}}$ or ${\displaystyle \operatorname {int} D^{n}}$.