co.combinatorics - Do all correlation coefficients induce a pseudometric?

The statement is false. Consider the Pearson sample correlation coefficient:

c(x,y) = (x-mean(x)).(y-mean(y))/sqrt(|x-mean(x)|^2*|y-mean(y)|^2)

Here is an example, where the triangle inequality for the distance defined in
the question is not satisfied

x = [0.5847 -0.3048 -0.4431 0.5032 -0.3401],

y = [0.2018 0.4547 -0.2230 0.3350 -0.7685],

z = [0.5226 -0.5159 -0.5439 0.3701 0.1671].