High-Dimensional Volume and Concentration - Sphere

1. Volume of a p-Dimensional Unit Sphere

Definition

The p-dimensional unit sphere Sp is defined as

Volume Derivation

The volume Vp is computed recursively using polar coordinates and the Beta function:

1. Recursive Integral

1. Beta Function Substitution:

1. Final Formula

2. Concentration of Volume Near the Equator

Spherical Caps and Volume Ratio

For a spherical cap at distance from the equator ( x1=0 ), the volume fraction is

Asymptotic Analysis

For large p, approximate

Key Result

For

Interpretation: Over 90% of the volume lies within of the equator in high dimensions.

3. Concentration in an Annulus at the Boundary

Volume of an Annulus

The annulus has volume

Exponential Decay

For small

Interpretation: For , nearly all volume concentrates in a thin shell of thickness