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?



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]. 


