Unit sphere

In Euclidean space of ⁠${\displaystyle n}$⁠ dimensions, the ⁠${\displaystyle (n-1)}$⁠-dimensional unit sphere is the set of all points ${\displaystyle (x_{1},\ldots ,x_{n})}$ which satisfy the equation $${\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=1.}$$

The open unit ⁠${\displaystyle n}$⁠-ball is the set of all points satisfying the inequality $${\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}<1,}$$ and closed unit ⁠${\displaystyle n}$⁠-ball is the set of all points satisfying the inequality $${\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\leq 1.}$$

The classical equation of a unit sphere is that of the ellipsoid with a radius of 1 and no alterations to the ⁠${\displaystyle x}$⁠-, ⁠${\displaystyle y}$⁠-, or ⁠${\displaystyle z}$⁠- axes: $${\displaystyle x^{2}+y^{2}+z^{2}=1}$$

The volume of the unit ball in Euclidean ⁠${\displaystyle n}$⁠-space, and the surface area of the unit sphere, appear in many important formulas of analysis. The volume of the unit ⁠${\displaystyle n}$⁠-ball, which we denote ${\displaystyle V_{n},}$ can be expressed by making use of the gamma function. It is $${\displaystyle V_{n}={\frac {\pi ^{n/2}}{\Gamma (1+n/2)}}={\begin{cases}{\pi ^{n/2}}/{(n/2)!}&\mathrm {if~} n\geq 0\mathrm {~is~even} \\[6mu]{2(2\pi )^{(n-1)/2}}/{n!!}&\mathrm {if~} n\geq 0\mathrm {~is~odd,} \end{cases}}}$$ where ${\displaystyle n!!}$ is the double factorial.

The hypervolume of the ⁠${\displaystyle (n-1)}$⁠-dimensional unit sphere (i.e., the "area" of the boundary of the ⁠${\displaystyle n}$⁠-dimensional unit ball), which we denote ${\displaystyle A_{n-1},}$ can be expressed as $${\displaystyle A_{n-1}=nV_{n}={\frac {n\pi ^{n/2}}{\Gamma (1+n/2)}}={\frac {2\pi ^{n/2}}{\Gamma (n/2)}}={\begin{cases}{2\pi ^{n/2}}/{(n/2-1)!}&\mathrm {if~} n\geq 1\mathrm {~is~even} \\[6mu]{2(2\pi )^{(n-1)/2}}/{(n-2)!!}&\mathrm {if~} n\geq 1\mathrm {~is~odd.} \end{cases}}}$$ For example, ${\displaystyle A_{0}=2}$ is the "area" of the boundary of the unit ball ${\displaystyle [-1,1]\subset \mathbb {R} }$, which simply counts the two points. Then ${\displaystyle A_{1}=2\pi }$ is the "area" of the boundary of the unit disc, which is the circumference of the unit circle. ${\displaystyle A_{2}=4\pi }$ is the area of the boundary of the unit ball ${\displaystyle \{x\in \mathbb {R} ^{3}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\leq 1\}}$, which is the surface area of the unit sphere ${\displaystyle \{x\in \mathbb {R} ^{3}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\}}$.

The surface areas and the volumes for some values of ⁠${\displaystyle n}$⁠ are as follows:

⁠${\displaystyle n}$⁠	${\displaystyle A_{n-1}}$ (surface area)		${\displaystyle V_{n}}$ (volume)
0			${\displaystyle (1/0!)\pi ^{0}}$	1
1	${\displaystyle 1(2^{1}/1!!)\pi ^{0}}$	2	${\displaystyle (2^{1}/1!!)\pi ^{0}}$	2
2	${\displaystyle 2(1/1!)\pi ^{1}=2\pi }$	6.283	${\displaystyle (1/1!)\pi ^{1}=\pi }$	3.141
3	${\displaystyle 3(2^{2}/3!!)\pi ^{1}=4\pi }$	12.57	${\displaystyle (2^{2}/3!!)\pi ^{1}=(4/3)\pi }$	4.189
4	${\displaystyle 4(1/2!)\pi ^{2}=2\pi ^{2}}$	19.74	${\displaystyle (1/2!)\pi ^{2}=(1/2)\pi ^{2}}$	4.935
5	${\displaystyle 5(2^{3}/5!!)\pi ^{2}=(8/3)\pi ^{2}}$	26.32	${\displaystyle (2^{3}/5!!)\pi ^{2}=(8/15)\pi ^{2}}$	5.264
6	${\displaystyle 6(1/3!)\pi ^{3}=\pi ^{3}}$	31.01	${\displaystyle (1/3!)\pi ^{3}=(1/6)\pi ^{3}}$	5.168
7	${\displaystyle 7(2^{4}/7!!)\pi ^{3}=(16/15)\pi ^{3}}$	33.07	${\displaystyle (2^{4}/7!!)\pi ^{3}=(16/105)\pi ^{3}}$	4.725
8	${\displaystyle 8(1/4!)\pi ^{4}=(1/3)\pi ^{4}}$	32.47	${\displaystyle (1/4!)\pi ^{4}=(1/24)\pi ^{4}}$	4.059
9	${\displaystyle 9(2^{5}/9!!)\pi ^{4}=(32/105)\pi ^{4}}$	29.69	${\displaystyle (2^{5}/9!!)\pi ^{4}=(32/945)\pi ^{4}}$	3.299
10	${\displaystyle 10(1/5!)\pi ^{5}=(1/12)\pi ^{5}}$	25.50	${\displaystyle (1/5!)\pi ^{5}=(1/120)\pi ^{5}}$	2.550

where the decimal expanded values for ${\displaystyle n\geq 2}$ are rounded to the displayed precision.

The ${\displaystyle A_{n}}$ values satisfy the recursion: $${\displaystyle A_{0}=2}$$ $${\displaystyle A_{1}=2\pi }$$ $${\displaystyle A_{n}={\frac {2\pi }{n-1}}A_{n-2}}$$ for ⁠${\displaystyle n>1}$⁠.

The ${\displaystyle V_{n}}$ values satisfy the recursion: $${\displaystyle V_{0}=1}$$ $${\displaystyle V_{1}=2}$$ $${\displaystyle V_{n}={\frac {2\pi }{n}}V_{n-2}}$$ for ⁠${\displaystyle n>1}$⁠.

The value ${\textstyle 2^{-n}V_{n}=\pi ^{n/2}{\big /}\,2^{n}\Gamma {\bigl (}1+{\tfrac {1}{2}}n{\bigr )}}$ at non-negative real values of ⁠${\displaystyle n}$⁠ is sometimes used for normalization of Hausdorff measure.^[1][2]

The surface area of an ⁠${\displaystyle (n-1)}$⁠-sphere with radius ⁠${\displaystyle r}$⁠ is ${\displaystyle A_{n-1}r^{n-1}}$ and the volume of an ⁠${\displaystyle n}$⁠-ball with radius ⁠${\displaystyle r}$⁠ is ${\displaystyle V_{n}r^{n}.}$ For instance, the area is ${\displaystyle A_{2}=4\pi r^{2}}$ for the two-dimensional surface of the three-dimensional ball of radius ${\displaystyle r.}$ The volume is ${\displaystyle V_{3}={\tfrac {4}{3}}\pi r^{3}}$ for the three-dimensional ball of radius ⁠${\displaystyle r}$⁠.

The open unit ball of a normed vector space ⁠${\displaystyle V}$⁠ with the norm ${\displaystyle \|\cdot \|}$ is given by $${\displaystyle \{x\in V:\|x\|<1\}}$$

It is the topological interior of the closed unit ball of ${\displaystyle (V,\|\cdot \|)\colon }$ $${\displaystyle \{x\in V:\|x\|\leq 1\}}$$

The latter is the disjoint union of the former and their common border, the unit sphere of ${\displaystyle (V,\|\cdot \|)\colon }$ $${\displaystyle \{x\in V:\|x\|=1\}}$$

The "shape" of the unit ball is entirely dependent on the chosen norm; it may well have "corners", and for example may look like ${\displaystyle [-1,1]^{n}}$ in the case of the max-norm in ⁠${\displaystyle \mathbb {R} ^{n}}$⁠. One obtains a naturally round ball as the unit ball pertaining to the usual Hilbert space norm, based in the finite-dimensional case on the Euclidean distance; its boundary is what is usually meant by the unit sphere.

Let ${\displaystyle x=(x_{1},...x_{n})\in \mathbb {R} ^{n}.}$ Define the usual ⁠${\displaystyle \ell _{p}}$⁠-norm for ${\displaystyle p\geq 1}$ as: $${\displaystyle \|x\|_{p}={\biggl (}\sum _{k=1}^{n}|x_{k}|^{p}{\biggr )}^{1/p}}$$

Then ${\displaystyle \|x\|_{2}}$ is the usual Hilbert space norm. ${\displaystyle \|x\|_{1}}$ is called the Hamming norm, or ⁠${\displaystyle \ell _{1}}$⁠-norm. The condition ${\displaystyle p\geq 1}$ is necessary in the definition of the ${\displaystyle \ell _{p}}$ norm, as the unit ball in any normed space must be convex as a consequence of the triangle inequality. Let ${\displaystyle \|x\|_{\infty }}$ denote the max-norm or ⁠${\displaystyle \ell _{\infty }}$⁠-norm of ⁠${\displaystyle x}$⁠.

Note that for the one-dimensional circumferences ${\displaystyle C_{p}}$ of the two-dimensional unit balls, we have: $${\displaystyle C_{1}=4{\sqrt {2}}}$$ is the minimum value. $${\displaystyle C_{2}=2\pi }$$ $${\displaystyle C_{\infty }=8}$$ is the maximum value.

All three of the above definitions can be straightforwardly generalized to a metric space, with respect to a chosen origin. However, topological considerations (interior, closure, border) need not apply in the same way (e.g., in ultrametric spaces, all of the three are simultaneously open and closed sets), and the unit sphere may even be empty in some metric spaces.

If ⁠${\displaystyle V}$⁠ is a linear space with a real quadratic form ${\displaystyle F:V\to \mathbb {R} ,}$ then ${\displaystyle \{p\in V:F(p)=1\}}$ may be called the unit sphere^[3][4] or unit quasi-sphere of ${\displaystyle V.}$ For example, the quadratic form ⁠${\displaystyle x^{2}-y^{2}}$⁠, when set equal to one, produces the unit hyperbola, which plays the role of the "unit circle" in the plane of split-complex numbers. Similarly, the quadratic form ${\displaystyle x^{2}}$ yields a pair of lines for the unit sphere in the dual number plane.

• Ball
• ⁠${\displaystyle n}$⁠-sphere
• Sphere
• Superellipse
• Unit circle
• Unit disk
• Unit tangent bundle
• Unit square