Let $u \in \mathbb{R}^n$ be a unit vector: $\|u\| = 1$. Is there a formula for the smallest $C > 0$ such that $|u_i| \geq C$ for some component $u_i$ where $i = 1,...,n$?
For example, if $u \in \mathbb{R}^2$ satisfying $u_1^2 + u_2^2 = 1$, then I believe $C = \frac{\sqrt{2}}{2}$ is the smallest constant such that either $|u_1| \geq C$ or $|u_2| \geq C$.
Your guess is right. Actually, for general $n$, when the sum of $u_i^2$ is 1, at least one of them should be larger than or equal to $1/n$. Otherwise, all of them is smaller tham $1/n$ and the sum can’t be $1$.