Reverse Triangle Inequality and other useful tidbits

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):

  1. |x+y| \leq |x| + |y|
  2. |x+y| \geq |x| - |y| \iff |x+y| +|y| \geq |x|
  3. |x+y| \geq |y| - |x| \iff|x+y| +|x| \geq |y|
  4. |x-y| \leq |x| + |y|
  5. |x-y| \geq |x| - |y| \iff |x-y| + |y| \geq |x|
  6. |x-y| \geq |y| - |x| \iff |x-y| + |x| \geq |y|

On the left hand side (of the \iff) 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 x, y, x+y. Inequalities 4, 5, 6 shows this intuition with the triangle described by the vectors x, y, x-y. The right hand side of the inequalities are arranged to show this explicitly.

Other useful tidbits:

  1. Let x \in \mathbb{R}^d, x = \begin{bmatrix} w \\ v \end{bmatrix}, w \in\mathbb{R}^n, v \in\mathbb{R}^m, n+m=d. Then \|x\|_2 \leq \|w\|_2 + \|v\|_2.
  2. Let x \in \mathbb{R}^d. If |x_i| \leq C, then \|x\|_2 \leq \sqrt{d} C.
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