In the book Computational Optimal Transport, the authors give the following definition for positive and negative definite functions:
Def 1. A symmetric function $k$ defined on a set $\mathcal X \times \mathcal X$ is said positive definite if for any $n \geq 0$, the familiy $x_1,...,x_n \in \mathcal X$ and vector $r \in \mathbb R^n$ the following inequality holds: $$ \sum^n_{i,j = 1} r_i r_j k(x_i,x_j) \geq 0 $$ The negative definite is just the opposite inequality. Finally, we call $k$ conditionally positive if this inequality if valid only for $r$ such that $\sum_i r_i = 0$.
Then the authors claim that the Euclidean distance is negative definite. My question is, how does one prove that indeed it is negative definite? I can perhaps prove that it is conditionally negative, but I don't see how to prove the actual claim.