TITLE: Is there any geometry where the triangle inequality fails?
QUESTION [27 upvotes]: We know from elementary school that the triangle inequality holds in Euclidean geometry. Some where in High School or in Univ., we come across non-Euclidean geometries (hyperbolic and Riemannian) and Absolute geometry where in both the inequality holds.
I am curious whether the triangle inequality is made to hold in any geometry( from the beginning) or is  a consequence of some axioms. Presumably, the denial of the inequality would create havoc in that conceivable geometry.
Thanks.

REPLY [8 votes]: The reverse triangle inequality|x+y|>|x|+|y| holds in the Minkowski Space of Special Relativity for two timelike vectors in the same direction. So geometries that deny the triangle inequality in some signficant way can be very important I guess.