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In mathematics — specifically, in geometric measure theory — spherical measure σ is the "natural" Borel measure on the n-sphere S. Spherical measure is often normalized so that it is a probability measure on the sphere, i.e. so that σ(S) = 1.

Definition of spherical measure

There are several ways to define spherical measure. One way is to use the usual "round" or "arclength" metric ρ_n on S; that is, for points x and y in S, ρ_n(x, y) is defined to be the (Euclidean) angle that they subtend at the centre of the sphere (the origin of R). Now construct n-dimensional Hausdorff measure H on the metric space (S, ρ_n) and define

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

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

Another method uses Lebesgue measure λ on the ambient Euclidean space R: for any measurable subset A of S, define σ(A) to be the (n + 1)-dimensional volume of the "wedge" in the ball B that it subtends at the origin. That is,

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

where

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

The fact that all these methods define the same measure on S follows from an elegant result of Christensen: all these measures are obviously uniformly distributed on S, 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 σ'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 R and let θ denote its normalized Haar measure (so that θ(O(n)) = 1). The orthogonal group also acts on the sphere S. Then, for any x ∈ S and any A ⊆ S,

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

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

Isoperimetric inequality

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

If A ⊆ S is any Borel set and B⊆ S is a ρ_n-ball with the same σ-measure as A, then, for any r > 0,

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

where A_r 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 σ(A) ≥ ⁠1/2⁠ 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. MR0260979
• Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9. MR1102015 (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. MR1333890 (See chapter 3)

• Measures (measure theory)