How to prove cosine distance does not satisfy triangle inequality metric in Euclidean space

Question

I am trying yo prove cosine distance doesn't satisfy metric rule

i know $$cosine \ distance = D(x,y)=1−S(x,y) = 1- \frac{x⋅y}{||x||||y||}$$

and able to proof cosine distance indeed satisfy other 3 matric rules $$dc(d1,d2)⩾0$$ $$dc(d1,d1)=0 $$ $$dc(d1,d2)=dc(d2,d1)$$ But I am wondering in what circumstance will cosine distance breaks triangle inequality metric? rule

Answer 1

That's only a distance if you restrict it to a sphere centered at the origin. Otherwise, any two points that lie on the same half-line starting at the origin will have zero distance. This is not compatible with the axioms of distance; no distinct point can have zero distance.