Given three vectors a, b, and c, is it possible that correlations between a and b, a and c, and b and c are all negative? I.e. is this possible?
It is possible if the size of the vector is 3 or larger. For example
The correlations are
We can prove that for vectors of size 2 this is not possible:
The formula makes sense: if a1 is larger than a2, b2 has to be larger than b1 to make the correlation negative.
Similarly for correlations between (a,c) and (b,c) we get
Clearly, all of these three formulas can not hold at the same time.