De Sitter–Schwarzschild metric

In mathematics — specifically, in geometric measure theory — spherical 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 “arclength” metric ρ_n on Sⁿ; that is, for points x and y in Sⁿ, ρ_n(x, y) 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 (Sⁿ, ρ_n) and define

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

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,

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

where

$\alpha (m):={\lambda }^m({\mathbf{B}}_1^m(0)).$

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,

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

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 ρ_n-ball with the same σⁿ-measure as A, then, for any r > 0,

${\sigma }^n(A_r)\ge {\sigma }^n(B_r),$

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

$A_r:=\{x\in {\mathbf{S}}^n\mid {\rho }_n(x,A)\le r\}.$

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

${\sigma }^n(A_r)\ge 1-\sqrt{\frac{\pi }{8}}\exp (-\frac{(n-1)r^2}{2}).$

References

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