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 $b=\sqrt{1-a^2\lfloor \frac{1}{a^2} \rfloor}$.



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> QUESTION 1: Is there a standard reference for facts like this?
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> QUESTION 2: What happes when we add the constraint $x\_{i} \geq b>0$?
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An elementary inequality: reference request
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>


