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The sparseness of a vector is defined a follows: $$\psi(\textbf{x}) = \frac{\sqrt{n}-\frac{\left(\sum_{i} x_i \right)}{\sqrt{\sum_{i} x_{i}^{2}}}}{\sqrt{n}-1}$$

So $\psi(\textbf{x}) =0 $ if $\sqrt{n} = \frac{\left(\sum_{i}x_i \right)}{\sqrt{\sum_{i} x_{i}^{2}}}$ or if $$\sqrt{n} \sqrt{\sum_{i} x_{i}^{2}} = \sum_{i} x_i$$

How does imply that all the $x_i$ are equal? Cauchy-Schwarz?

Likewise $\psi(\textbf{x}) = 1$ iff $\textbf{x}$ contains a single non-zero element. How does this follow? We know that $$\sqrt{n}-\frac{\left(\sum_{i} x_i \right)}{\sqrt{\sum_{i} x_{i}^{2}}} = \sqrt{n}-1$$

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