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. 



Is there any geometry where the triangle inequality fails?
There are people who seriously study quasi-normed spaces. The most natural examples are $\ell\_p$ spaces for p strictly between 0 and 1 (the "norm" given by the usual formula and the distance given by the norm of the difference). Although these spaces do not satisfy the triangle inequality, you get an inequality of the form $\|x+y\|\leq C(\|x\|+\|y\|)$.


