Relation between l1-l2 norms

[Firdaus Janoos]

Hello,

Does anybody know of a result about the expected L1 norm of a n-dim random vector sampled uniformly on the unit sphere ?

I.e. if U = {U_1 ... U_n} is sampled uniformly on the unit sphere, i.e. |U|_2 = 1, then what is E[ |U|_1 ] ?

Thanks.
-firdaus

[Firdaus Janoos]

Just to clarify, these are two equivalent ways of posing this question:

a) If U = {U_1 ... U_n} is sampled uniformly on the unit sphere, i.e. |U|_2 = 1, then what is E[ |U|_1 ] for arbitrary 'n'.

b) With x \in R^n what is \int_{ ||x||_2=1 } ||x||_1 dx for arbitrary 'n'.

The approach I'm currently use is to compute \int ||x||_1 dx over the positive orthant of the n-dim unit l2 ball (||x||_2 = 1) or vice versa (i.e. \int ||x||_2 dx over the positive orthant of the n-dim l1 unit ball - which is easier to do).

The draw back is that I can evaluate this integral for a specific 'n' and get a numerical value for each 'n'. But I'm interested in a general (closed form) formula that would give me this for *any* n.

Any pointers would be appreciated.

Thanks!