I've been wondering, is there a generalized version of the triangle inequality that is useful in math? I recently saw the definition of a metric space, and wondered what would happen if you want a function that satisfies the first two conditions of a metric function but instead of the third condition, satisfies something like $d(x_{1}, x_{n}) \leq d(x_{1}, x_{2}) + ... + d(x_{n-1}, x_{n}))$? Does an inequality like this show up anywhere else in math? If not, why is it that having this generalized inequality with $n = 3$ so special? I've looked at the polygon example of the wiki article: http://en.wikipedia.org/wiki/Triangle_inequality#Generalization_of_the_inequality_to_any_polygon and tried reading http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_35.pdf but that's above my level right now. Any help/advice is appreciated. Thanks!
Sincerely,
Vien