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If $x_1, x_2 \in \mathbb{C}^n$, then is there any lower and upper bound for $\|x_1 + x_2\|_{\infty}$, where $\| x\|_{\infty} := \max_{i=1,\ldots,n} |x_i| $?

More specifically, I am wondering, is there a lower and upper bound in this below format $\alpha_1 \|x_1\|_{\infty} + \alpha_2 \|x_2\|_{\infty} \leq \|x_1 + x_2\|_{\infty} \leq \beta_1 \|x_1\|_{\infty} + \beta_2 \|x_2\|_{\infty}$? If yes, then what can be $\alpha_1 \in \mathbb{R}$, $\alpha_2 \in \mathbb{R}$, $\beta_1 \in \mathbb{R}$, and $\beta_2 \in \mathbb{R}$?

EDIT: For upper bound, the triangle inequality can be invoked, i.e., $ \|x_1 + x_2\|_{\infty} \leq \beta_1 \|x_1\|_{\infty} + \beta_2 \|x_2\|_{\infty}$, where $\beta_1 = 1$ and $\beta_2 = 1$.

EDIT: For lower bound, one stupid option can be taking the difference such that $\| x_1\|_{\infty} - \| x_2\|_{\infty} \leq \| x_1 + x_2\|_{\infty} $ assuming $\| x_1\|_{\infty} \geq \| x_2\|_{\infty} $. Not sure if this will result into the tight bounds

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  • $\begingroup$ Are you familiar with the triangle inequality? $\endgroup$ May 15 at 13:56
  • $\begingroup$ @TMGallagher Yes, true. That can give me an upper bound. How about the lower bound? $\endgroup$
    – learning
    May 15 at 13:58
  • $\begingroup$ What have you tried toward seeing if a non-zero lower bound is possible? $\endgroup$ May 15 at 14:00
  • $\begingroup$ @TMGallagher One stupid option can be taking the difference such that $\| x_1\|_{\infty} - \| x_2\|_{\infty} \leq \| x_1 + x_2\|_{\infty} $ assuming $\| x_1\|_{\infty} \geq \| x_2\|_{\infty} $. Not sure if this will result into the tight bounds $\endgroup$
    – learning
    May 15 at 14:04
  • $\begingroup$ I definitely wouldn't call that ``stupid,'' and you can investigate whether it's a tight lower bound. $\endgroup$ May 15 at 14:16

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