Originally Posted by **hoeltgman**

I've got a few things that don't seem really clear to me. The first point is clear. I derivate and find out that it's always positive....

I do have problems to see how come from your second point to my original problem. If I say a = |x-y| then b could be |x-z| or |y-z|. Something is missing here I think. I tried your way and ended up with a + b + 1 -ab < 0. which doesn't really help since even if I transform back, I find can find the third distance, but I still have a term which keeps me from getting any good conclusion.