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Consider the problem of minimizing $\sum_{i=1}^{n}{x_{i}}$ under the constraints $\sum_{i=1}^{n}{x_{i}^{2}}=1$ and $x_{i} \geq 0$. Obviously the solution is given by the vector $(1,0,\ldots,0)$.

Now consider the variant where we add the constraint $x_{i} \leq a$. I am quite sure that the smallest sum is attained by the vector $(a,a,\ldots,a,b,0,\ldots,0)$ with $a$ repeated $\lfloor \frac{1}{a^2} \rfloor$ times and $b=\sqrt{1-a^2\lfloor \frac{1}{a^2} \rfloor}$.

QUESTION 1: Is there a standard reference for facts like this?

QUESTION 2: What happes when we add the constraint $x_{i} \geq b>0$?

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    $\begingroup$ This is an easy consequence of Karamata's inequality (aka Hardy-Littlewood-Polya, aka Majorization) applied to the function $\sqrt{x}$. $\endgroup$
    – zeb
    Commented Feb 8, 2014 at 1:15
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    $\begingroup$ @zeb Thanks! This is just what I was looking for. Perhaps you should turn the comment into an answer, so that I can accept it. $\endgroup$ Commented Feb 8, 2014 at 1:38

1 Answer 1

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Karamata's paper: http://elib.mi.sanu.ac.rs/files/journals/publ/1/11.pdf

A few related papers are listed out in this AoPS forum post: http://www.artofproblemsolving.com/Forum/viewtopic.php?p=123878&#p123878

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