The proof technique for “How often do the L1 and L2 norms agree?” used as a lemma a characterization of the L1 norm of n-dimensional vectors chosen uniformly with L2 norm equal to 1.
For a n-dimensional vector with unit L2 norm we can see L1 norms as small as 1 (for the
(1, 0, ..., 0) vector), or as large as
sqrt(n) (for the
(sqrt(n), ..., sqrt(n)) vector). Some of the situations are indicated in the following diagram.
What we were able to prove here is that for large
n the expected L1 norm approaches
sqrt(2 n / π) (within a constant multiple of the maximum possible) and the variance of this is approaching
1 - 3 / π.
The constant variance means this distribution is tightly concentrated around its mean.
Kind of a cool fact to know.