How many coordinates we need to remove from a uniformly selected point on the unit sphere before the remaining are on the same order of magnitude?

Solution 1:

Unfortunately, your ideal upper bounds on $Y$ cannot hold. Here is one easy way to see this. Sample a standard Gaussian $g$ in $\mathbb R^d$. By rotational invariance, $g/\| g\|$ is a uniform vector on the sphere. It also turns out that $\| g\|$ is tightly concentrated around $\sqrt{d}$. There are several ways to see this, for example this follows from Gaussian concentration.

But for a standard Gaussian vector, the number of coordinates larger than a constant is simply a binomial with parameters $d$ and $p=\Omega(1)$. For sure, there are at least, say, $dp/100$ such coordinates with high probability. If you adjust all constants properly, this should imply that $Y$ has $\Omega(d)$ coordinates larger than $c/\sqrt{d}$ with high probability.