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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 $|u_2| \geq C$.

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  • $\begingroup$ This is basically the pigeon hole principle. $\endgroup$
    – copper.hat
    Commented Jun 25, 2021 at 0:54
  • $\begingroup$ Can you elaborate? $\endgroup$
    – Frederic Chopin
    Commented Jun 25, 2021 at 1:28
  • $\begingroup$ If you have $n$ boxes (the components) with contents $u_k^2$ then at least one of the boxes must be $\ge {\sum_k u_k^2 \over n}$. (Otherwise an easy contradiction.) $\endgroup$
    – copper.hat
    Commented Jun 25, 2021 at 1:56

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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 $1$.

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