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 $x_{i} \geq b>0$?
