Suppose $X_k$ is a matrix made by stacking $k$ random vectors of unit norm sampled from some sufficiently nice $d$-dimensional distribution.
How would I show that the following quantity is monotonically increasing in $k$ for operator norm $\|\cdot\|$?
$$E[(\|X_k\|-1)^2]/k$$
Expecting it to be true because when $k$ is small relative to $d$, a random vector has high probability to be orthogonal to existing ones and may not affect the norm much.
Simple simulation
