# Spherical measure

In mathematics — specifically, in geometric measure theoryspherical measure σn is the “natural” Borel measure on the n-sphere Sn. Spherical measure is often normalized so that it is a probability measure on the sphere, i.e. so that σn(Sn) = 1.

## Definition of spherical measure

There are several ways to define spherical measure. One way is to use the usual “round” or “arclengthmetric ρn on Sn; that is, for points x and y in Sn, ρn(xy) is defined to be the (Euclidean) angle that they subtend at the centre of the sphere (the origin of Rn+1). Now construct n-dimensional Hausdorff measure Hn on the metric space (Snρn) and define

${\displaystyle \sigma ^{n}={\frac {1}{H^{n}(\mathbf {S} ^{n})}}H^{n}.}$

One could also have given Sn the metric that it inherits as a subspace of the Euclidean space Rn+1; the same spherical measure results from this choice of metric.

Another method uses Lebesgue measure λn+1 on the ambient Euclidean space Rn+1: for any measurable subset A of Sn, define σn(A) to be the (n + 1)-dimensional volume of the “wedge” in the ball Bn+1 that it subtends at the origin. That is,

${\displaystyle \sigma ^{n}(A):={\frac {1}{\alpha (n+1)}}\lambda ^{n+1}(\{tx\mid x\in A,t\in [0,1]\}),}$

where

${\displaystyle \alpha (m):=\lambda ^{m}(\mathbf {B} _{1}^{m}(0)).}$

The fact that all these methods define the same measure on Sn follows from an elegant result of Christensen: all these measures are obviously uniformly distributed on Sn, and any two uniformly distributed Borel regular measures on a separable metric space must be constant (positive) multiples of one another. Since all our candidate σn’s have been normalized to be probability measures, they are all the same measure.

## Relationship with other measures

The relationship of spherical measure to Hausdorff measure on the sphere and Lebesgue measure on the ambient space has already been discussed.

Spherical measure has a nice relationship to Haar measure on the orthogonal group. Let O(n) denote the orthogonal group acting on Rn and let θn denote its normalized Haar measure (so that θn(O(n)) = 1). The orthogonal group also acts on the sphere Sn−1. Then, for any x ∈ Sn−1 and any A ⊆ Sn−1,

${\displaystyle \theta ^{n}(\{g\in \mathrm {O} (n)\mid g(x)\in A\})=\sigma ^{n-1}(A).}$

In the case that Sn is a topological group (that is, when n is 0, 1 or 3), spherical measure σn coincides with (normalized) Haar measure on Sn.

## Isoperimetric inequality

There is an isoperimetric inequality for the sphere with its usual metric and spherical measure (see Ledoux & Talagrand, chapter 1):

If A ⊆ Sn−1 is any Borel set and B⊆ Sn−1 is a ρn-ball with the same σn-measure as A, then, for any r > 0,

${\displaystyle \sigma ^{n}(A_{r})\geq \sigma ^{n}(B_{r}),}$

where Ar denotes the “inflation” of A by r, i.e.

${\displaystyle A_{r}:=\{x\in \mathbf {S} ^{n}\mid \rho _{n}(x,A)\leq r\}.}$

In particular, if σn(A) ≥ ½ and n ≥ 2, then

${\displaystyle \sigma ^{n}(A_{r})\geq 1-{\sqrt {\frac {\pi }{8}}}\,\exp \left(-{\frac {(n-1)r^{2}}{2}}\right).}$

## References

• Christensen, Jens Peter Reus (1970). "On some measures analogous to Haar measure". Mathematica Scandinavica. 26: 103–106. ISSN 0025-5521. MR 0260979
• Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9. MR 1102015 (See chapter 1)
• Mattila, Pertti (1995). Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability. Cambridge Studies in Advanced Mathematics No. 44. Cambridge: Cambridge University Press. pp. xii+343. ISBN 0-521-46576-1. MR 1333890 (See chapter 3)