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Suppose I am told I have N numbers and that the sum of the square of those numbers is S. Can anything be said about the sum of those original N numbers?

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Cauchy-Schwarz gives $$\sum_{n=1}^N |x_n| \le \sqrt{N} \sqrt{\sum_{n=1}^N x_n^2}.$$

For a lower bound, we have $$\left(\sum_{n=1}^N |x_n|\right)^2 \ge \sum_{n=1}^N x_n^2.$$

So, $\sqrt{S} \le \sum_{n=1}^N |x_n| \le \sqrt{NS}$.

Without further information, these bounds cannot be improved. If all of the numbers are the same, then $\sum_{n=1}^N |x_n| = \sqrt{NS}$. If all of the numbers are zero except one of them, then $\sum_{n=1}^N |x_n| = \sqrt{S}$.

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