Significance of n-sphere in statistical physics

[sunrah]

Homework Statement

Just trying to understand something my instructor said. For large n, the volume of the sphere exists in a thin shell at its surface. He asked us what relationship this has to statistical physics.

Homework Equations

The Attempt at a Solution

The only thing I can think of is at very high numbers of dimension the volume of a sphere is akin to its surface area ??

 

[mfb]

Imagine N particles with velocity v_i with a fixed total energy E. How does their phase space look like?
How does the phase space (and its volume) look like if you reduce E a tiny bit?