Condition for inequality of sum equation and sum For given two sets of $n$ numbers satisfying:
$$x_1^2+x_2^2+\cdots+x_n^2 ≥ y_1^2+y_2^2+\cdots+y_n^2$$

Here ($x_i^2$ presents the square of $x_i$) and
$x_i, y_i$ are not negative numbers.

What is the condition of $x_i, y_i$ in which the following statement is true?
$$x_1+x_2+\cdots+x_n ≥ y_1+y_2+\cdots+y_n$$
One condition that comes right away is that $$\sum_{i\ne j}x_ix_j\ge \sum_{i\ne j}y_iy_j$$
$$x_1+x_2+\cdots+x_n\geq y_1+y_2+\cdots+y_n$$ need not be true.  For example, with $n=10$, take $x_1 = 6$ and for all $i&gt;1$ take $x_i=1$.  Then take all ten $y_k = 2$.

$$x_1^2 + x_2^2 + \cdots+ x_{10}^2 = 16 + 9\cdot 1 = 45 &gt; y_1^2 + y_2^2 + \cdots + y_{10}^2 = 10\cdot 4 =40 $$ 
$$x_1+x_2+\cdots+x_n = 6+9\cdot 1 = 15 &lt;  y_1+y_2 +\cdots+y_n = 10\cdot 2 = 20$$