Remembering the Reverse Triangle Inequality

21 Mar 2021 - Tags: quick-analysis-tricks

The quarter is over, and now that I’m vaccinated (twice!) I feel comfortable seeing people again. So I flew east coast to see my family and a bunch of friends. Before I left, I had a few ideas for blog posts, and figured I would get around to writing one now.

I’ve made it known that I struggle with analysis1, and one manifestation of this is a complete inability to remember elementary facts about inequalities. It took me a long time to feel comfortable with things as basic as “which way does the triangle inequality go?”, and until fairly recently things like Cauchy-Schwarz were almost entirely beyond me. Over the past year or two, I’ve been trying to answer lots of analysis questions on mse, as well as read lots of books on analysis and solve lots of problems, and (thankfully) some of it is starting to stick. But one inequality that I always seem to forget is the reverse triangle inequality:

$$\Bigg | |x| - |y| \Bigg | \leq |x-y|$$

I don’t know many ways for showing a lower bound on absolute values, but almost every time I need one, I go through the following process:

1. “Doesn’t the reverse triangle inequality give a lower bound?”
2. “I wonder if I should use that. Let me google it!”
3. “Oh right, that’s what it says. How do I always forget this?”
4. “This is actually not as useful as I would have liked. Oh well.”

The most recent time I went through this, something on the wikipedia page really clicked with me, and I’m not sure why it never clicked before. The geometric intuition2 never really stays in my head, but for some reason this did:

The reverse triangle inequality says that the norm $\lVert \cdot \rVert$ of some vector space $X$ is 1-lipschitz as a function from $X \to \mathbb{R}$:

$\Bigg | \lVert x \rVert - \lVert y \rVert \Bigg | \leq \lVert x-y \rVert$

Or, even more suggestively:

$$d_\mathbb{R}(\lVert x \rVert, \lVert y \rVert) \leq d_X(x,y)$$

I’m trying to see why this is more memorable for me, and moreover why it’s suddenly more memorable. Because I know that I’ve seen this before.

I think part of it is the visual and semantic distinction that we get by writing $\lVert \cdot \rVert$ instead of $|\cdot|$. When everything in sight was a real number, there were too many combinations of what we should and shouldn’t be absolute-value-ing. As with many things in math and computer science, taking some time to recognize the types involved in an equation or proof, and then making sure to distinguish these types inyour mind, helps a lot for keeping the structures straight.

I think another reason this is memorable is because the notion of lipschitz maps has become something I feel familiar with. When I was taking my first undergraduate analysis class, I really didn’t know why we should care about the various strengthenings of continuity. Over time I’ve learned to better appreciate their differences, and I feel like lipschitz-ness is one of the regularity conditions that I understand best3. It also makes intuitive sense that taking the norm of a vector should be a very regular operation.

Anyways, this isn’t so much a “trick” as a “mnemonic”, but I wanted to say something about it anyways because I think it would have helped younger me. At the very least, it was nice to write up a really short post with a somewhat obvious observation. To make it somewhat more worth your time, here’s a picture of my old cat Oreo. I got to visit her while I was visiting Remy in New York!

1. Though I’ve done fairly well the past two quarters, which has been a real confidence boost… It’s feeling better, but I still don’t feel like I understand things as well as I should, and while it’s coming faster, I still don’t feel like it’s coming naturally… Maybe it’s imposter syndrome? Who’s to say ¯_(ツ)_/¯

2. The difference between two legs of a triangle must be less than than the length of the third leg. Otherwise, by adding the length of the shorter leg to both sides you would violate the triangle inequality.

3. Not that that’s saying much.