Let $x=(x_1, \ldots, x_n)$ be a vector on $S^{n-1}$. Reorder coordinates such that $|x_1|\leq \ldots \leq |x_n|$.
I am wondering if there is a some relation between the absolute value of the coordinates of the vector $x$?
For example, to have all non-zero coordinates, I should have that, say, all of the coordinates would be $1/\sqrt n$. What happened if vector would have $k$ non-zero coordinates--would be the upper bound for $x_i$?
Thank you.
Even if all the coordinates are non-zero, one could be almost $1$ and the rest could be very small. You can say that at least one coordinate is at least $\frac 1{\sqrt n}$ in absolute value, but that is about it.