View source for Ball

A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015120/b0151201.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015120/b0151202.png" /> in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015120/b0151203.png" /> at a distance from a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015120/b0151204.png" /> (the centre of the ball) less than (an open ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015120/b0151205.png" />), or not greater than (a closed ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015120/b0151206.png" />) a quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015120/b0151207.png" /> (the radius of the ball), i.e.

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A ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015120/b0151209.png" /> is a line segment, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015120/b01512010.png" /> is a disc, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015120/b01512011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015120/b01512012.png" /> is sometimes called a hyperball. The boundary (surface) of a ball is a [[Sphere|sphere]].

The volume of a ball is

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015120/b01512014.png" /> is the surface of the boundary sphere and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015120/b01512015.png" /> is the gamma-function: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015120/b01512016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015120/b01512017.png" />. In particular,

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With the increase of the dimension, the volume of a ball  "concentrates"  at its surface:

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A ball is the simplest geometrical figure. Its topology is trivial. Among all bodies of an equal volume, a ball has minimal surface, and among all bodies of an equal surface, it has maximal volume.

In exactly the same manner a ball can be defined in a metric space; however, in this case it need not be, for example, strictly convex, its surface may have non-smooth points, etc., that is, it may have all phenomena characteristic of arbitrary convex bodies.

Unlike a finite-dimensional ball, an infinite-dimensional ball, being the direct limit of a sequence of balls of successive dimensions imbedded in one another, does not have a compact closure. On the contrary, the compactness of a ball in a topological vector space indicates the finite dimensionality of the latter.

For references, see [[Sphere|Sphere]].