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$x_1, ..., x_n \in \mathbb{R}$

Prove that $(max(0,1-|x_i-x_j|))_{i,j}$ is a positive semi-definite matrix.

Classic methods don't seem to work...

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  • $\begingroup$ Do you mean that its entries are non-negative or that the matrix is positive definite? $\endgroup$ – Omnomnomnom Mar 3 '15 at 20:01
  • $\begingroup$ Positive semi-definite. $\endgroup$ – Arnaud Mar 3 '15 at 20:01
  • $\begingroup$ I am wondering if, more generally, $\left(1-d\left(x_i,x_j\right)\right)_{i,j}$ is a positive-definite matrix whenever $x_1,x_2,\ldots,x_n$ are $n$ elements of a metric space $M$ with a distance function $d$. $\endgroup$ – darij grinberg Mar 3 '15 at 20:02

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