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If K is a pd kernel and you define d(x,y)=K(x,x)+K(y,y)-2K(x,y), how do you prove that d(x,y) is a distance? (specially the triangle inequality)

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    Have you tried anything – 2017-01-30
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    Yes, but I get stuck with the triangle inequality :/ – 2017-01-30

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Hints. Step 1. Write the triangle inequality in terms of $K$.
Step 2. Rephrase your inequality as $\text{sum of different $K$-values} \ge 0$.
Step 3. Use that $K$ is positive definite, i. e. wisely choose $c_x$, $c_y$, $c_z$ such that your inequality reads $$ \sum_{a,b \in \{x,y,z\}} c_a c_b K(a,b) \ge 0 $$