The reverse triangle inequality is one of those things that are simple, but always takes me a couple seconds to wrap my head around. So in this post, I list this inequality (for me and others to look on when those couple seconds are taking longer than they should) and also some other useful tidbits that I used to prove things in my internship at Microsoft this past summer.
Reverse Triangle Inequalities (and normal triangle inequalities):
On the left hand side (of the ) are the inequalities, and the right hand side are the same inequalities arranged in a different way. Recall the grade school intuition that for any valid triangle, the sum of the lengths of 2 sides is always greater than or equal to the 3rd side. Inequalities 1, 2, 3 shows this intuition with the triangle described by the vectors . Inequalities 4, 5, 6 shows this intuition with the triangle described by the vectors . The right hand side of the inequalities are arranged to show this explicitly.
Other useful tidbits:
- Let . Then .
- Let . If , then .