Given $n$ positive number that $x_1+x_2+...+x_n=k$. I want to minimize $\sum \sqrt{x_i^2+1}$. I guess that this sum is minmum when $x_1=x_2=...=x_n$, but I cant prove it. can anybody help me? Thanks.

  • $\begingroup$ Of course! my mistake. $\endgroup$ – uniquesolution Nov 19 '15 at 10:35

Note that $f(u) = \sqrt{u^2 +1} $ is convex. Using Jensen's inequality,

$$\frac{1}{n} \sum \sqrt{x_i^2 + 1} \ge \sqrt{\left(\frac{1}{n} \sum x_i\right)^2 +1} = \sqrt{\left(\frac{k}{n}\right)^2 + 1}$$

with equality at $x_i = \frac{k}{n} ~\forall i$.


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