Do all correlation coefficients induce a pseudometric?

Question

The Kendall tau distance was originally defined as a correlation coefficient. It seems clear to me that every metric function $d$ that is bounded by $b$, induces a correlation coefficient. That is:

Let $d$ be a metric. Then $-d(x,y)/b+1$ is a correlation coefficient.

I wonder if the converse is true, too:

Let $c(x,y)$ be a correlation coefficient. Then $-c(x,y) + 1$ is a pseudometric.

Is the latter statement true or false?

Answer 1

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].